[MAA 3.17-3.19] PLANES

INTERNATIONAL BACCALAUREATE Mathematics: analysis and approaches MAA EXERCISES [MAA 3.17-3.19] PLANES Compiled by Christos Nikolaidis O. 1. Practice questions [Maximum mark: 10] [without GDC] 1  2     Consider the point A(2, 1, 5) and the vectors b   0  and c   1  .  3  4     (a) Write down the vector equation of the plane Π that passes through the point A and is parallel to the vectors b and c . [2] (b) Find the vector b  c . [3] (c) The equation of the plane Π can be written in the form r  n  a  n . Write down (d) the vectors n and a . [2] Hence, find the Cartesian equation ax  by  cz  d of the plane Π . [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 1 [MAA 3.17-3.19] PLANES 2. [Maximum mark: 10] [without GDC] Consider the points A(2, 1,  5) , B(1, 1, 1) and C(3,  1,  2) . (a) Find the vectors AB  AC . [4] Hence, (b) find the Cartesian equation of the plane of the triangle ABC. [3] (c) find the area of the parallelogram defined by the vectors AB, AC . [2] (d) find the area of the triangle ABC. [1] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 2 [MAA 3.17-3.19] PLANES 3. [Maximum mark: 12] [with GDC] The point A (2, 5, –1) is on the line L , which is perpendicular to the plane with equation x  y  z 1  0 (a) Find the Cartesian equation of the line L . [2] (b) Find the point of intersection of the line L and the plane. [4] (c) The point A is reflected in the plane. Find the coordinates of the image of A. [2] (d) Calculate the distance from the point B(2, 0, 6) to the line L . [4] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 3 [MAA 3.17-3.19] PLANES 4. [Maximum mark: 21] [with GDC] Consider the lines 1  2     L1 : r   2     3   3 1      1  4     L2 : r  1    7   1  4     and the planes Π1 : 2x  3y  z  7 Π2 : 4 x  7 y  4 z  19 Find the following. (a) The acute angle between the lines L1 and L2 . [4] (b) The acute angle between the planes Π1 and Π2 . [1] (c) The acute angle between the line L1 and the plane Π 2 . [2] (d) The acute angle between the y -axis and the plane Π 2 . [3] (e) The point of intersection of the lines L1 and L2 . [5] (f) The point of intersection of the line L1 and plane Π1 . [3] (g) The line of intersection of the planes Π1 and Π2 . 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Page 4 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 5 [MAA 3.17-3.19] PLANES A. 5. Exam style questions (SHORT) [Maximum mark: 6] (a) (b) [without GDC] If u = i +2j + 3k and v = 2i – j + 2k, show that u × v = 7 i + 4 j – 5 k. [2] Let w = λu + μv where  and  are scalars. Show that w is perpendicular to the line of intersection of the planes x  2 y  3 z  5 and 2 x  y  2 z  7 for all values of  and  . [4] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 6. [Maximum mark: 6] [with GDC]  4    A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector  1    2   and meets the plane  : x  3 y  2 z  24  0 . Find the angle that the ray of light makes with the plane. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 6 [MAA 3.17-3.19] PLANES 7. [Maximum mark: 6] The line [with GDC] x 1 y  2 z  1 is reflected in the plane x  y  z  1 . Calculate the angle   1 2 2 between the line and its reflection. Give your answer in radians. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 8. [Maximum mark: 6] [with GDC] Find the cosine of the angle  between the planes  1 and  2 , where  1 has equation 2 x  y  z  2 and  2 has equation x  2 y  z  6 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 9. [Maximum mark: 6] [with GDC] Find the angle between the plane 3 x  2 y  4 z  12 and the z -axis. Give your answer to the nearest degree. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 7 [MAA 3.17-3.19] PLANES 10. [Maximum mark: 6] [without GDC] Consider the point A(1, −1, 4) and the line L2 with equation r = 2i+4j +7k + t(2i +j +3k) where t  . Find the Cartesian equation of the plane that contains both the line L2 and point A. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 11. [Maximum mark: 6] [without GDC] Find an equation of the plane containing the two lines x –1 1– y x 1 2 – y z  2 .  z – 2 and   2 3 3 5 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 8 [MAA 3.17-3.19] PLANES 12. [Maximum mark: 6] [without GDC]  2   x4 y2 z 6 A plane Π has equation r    1  16 and a line l has equations   2 4 1 1   Show that the line l lies in the plane. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 13. [Maximum mark: 5] [without GDC] The plane 6 x  2 y  z  11 contains the line x  1  y 1 z  3 . Find l .  2 l .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 9 [MAA 3.17-3.19] PLANES 14. [Maximum mark: 5] [without GDC] Find the coordinates of the point where the line given by the parametric equations x  2  4 , y    2 , z  3  2 , intersects the plane with equation 2 x  3 y  z  2 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 15. [Maximum mark: 5] [without GDC] Find the coordinates of the point of intersection of the line L with the plane P where: x  3 y –1 z –1   2 –1 2 P :2x  3y – z  – 5 L: .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 10 [MAA 3.17-3.19] PLANES 16. [Maximum mark: 6] [without GDC] The line r = i + k + μ(i – j + 2k) and the plane 2 x  y  z  2  0 intersect at the point P. Find the coordinates of P. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 17. [Maximum mark: 6] The line [without GDC] x3 5 z and the plane 2 x  y  3 z  10 intersect at the point P.  y 1  2 3 Find the coordinates of P. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 11 [MAA 3.17-3.19] PLANES 18. [Maximum mark: 5] [with / without GDC] Find the equation of the line of intersection of the two planes 4 x  y  z  2 and 3 x  y  2 z  1 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 19. [Maximum mark: 5 [with / without GDC] Find an equation for the line of intersection of the following two planes. x  2 y  3z  2 2x  3 y  5z  3 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 12 [MAA 3.17-3.19] PLANES 20. [Maximum mark: 6] [without GDC] The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane 2 x  y  z  6 . Find the coordinates of A. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 21. [Maximum mark: 6] [without GDC] Let P be the point (1, 0, – 2) and Π be the plane x  y  2 z  3  0 . Let P΄ be the reflection of P in the plane Π . Find the coordinates of the point P΄. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 13 [MAA 3.17-3.19] PLANES 22. [Maximum mark: 8] [without GDC] Three points A, B and C have coordinates (2, 1,–2), (2,–1,–1) and (1, 2, 2) respectively. The vectors OA , OB and OC , where O is the origin, form three concurrent edges of a parallelepiped OAPBCQSR as shown in the following diagram. P S A Q B R O C (a) Find the coordinates of P, Q, R and S. [4] (b) Find an equation for the plane OAPB. [2] (c) Calculate the volume, V, of the parallelepiped given that V = OA  OB  OC . [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 14 [MAA 3.17-3.19] PLANES INTERSECTION OF THREE PLANES 23. [Maximum mark: 22] [without GDC] Please revisit the exercises on paragraph [MAA 1.10] SYSTEMS OF LINEAR EQUATIONS Complete the table below. Geometric relationship of the three planes (state the vector equation of the line of intersection if applicable) [MAA 1.10] Exercise (i) The intersection point of the three planes is (1, 1,1) (ii) There is no point of intersection. The three planes form a prism 1 2 3 4 5 6 If k  3 : 7 If k  3 : If k  5 : 8 If k  5 : 1 3 If k  3, : 9 If k  1 : 3 If k  3 : If a  1 : 10 If a  1 : If k  1 : 11 0 1 0   1   (iii) The three planes intersect in the straight line: r   3     2      If k  1 : Page 15 [MAA 3.17-3.19] PLANES B. 24. Exam style questions (LONG) [Maximum mark: 12] (a) [with GDC] The point P(1, 2, 11) lies in the plane  1 . The vector 3i – 4 j + k is perpendicular to  1 . Find the Cartesian equation of  1 . (b) (c) [2] The plane  2 has equation x  3 y  z  4 . (i) Show that the point P also lies in the plane  2 . (ii) Find a vector equation of the line of intersection of  1 and  2 . Find the acute angle between  1 and  2 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 16 [5] [5] [MAA 3.17-3.19] PLANES 25. [Maximum mark: 11] [without GDC] The plane  contains the line x 1 y 1 z  5 and the point (1, 2,3) .   2 3 6 (a) Show that the equation of  is 6 x  2 y  3 z  7 . [7] (b) Calculate the distance of the plane  from the origin. [4] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 17 [MAA 3.17-3.19] PLANES 26. [Maximum mark: 13] [with GDC] A plane π1 has equation (a) 6   r   5   15 .  4   A point P ( p,  p, p ) lies on plane π1 and Q is the point where the plane π1 meets the y -axis. (i) Find the coordinates of P and of Q. (ii) Show that PQ is parallel to the vector u, where u = i – 2j + k. (b) [5] Another plane π2 intersects π1 in the line (PQ). The point T(1,0,–1) lies on π2 . (i) Find the equation of π2 , giving your answer in the form Ax+By+Cz = D. (ii) Find the angle between π1 and π2 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 18 [8] [MAA 3.17-3.19] PLANES 27. [Maximum mark: 13] (a) [with / without GDC] Solve the following system of linear equations x  3 y  2 z  6 2 x  y  3z  7 3x  y  z  6 [3] (b) Find the vector v = (i + 3j – 2k) × (2i + j + 3k). (c) If a = i + 3j – 2k, b = 2i + j + 3k and u = ma + nb where m, n are scalars, and u  0 , show that v is perpendicular to u for all m and n. (d) [3] [3] The line l lies in the plane 3 x  y  z  6 , passes through the point (1, –1, 2) and is perpendicular to v. Find the equation of l . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 19 [4] [MAA 3.17-3.19] PLANES 28. [Maximum mark: 22] [with GDC] The coordinates of the points P, Q, R and S are (4,1,–1), (3,3,5), (1,0, 2c), and (1,1,2), respectively. (a) Find the value of c so that the vectors QR and PR are orthogonal. [7] For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R . (b) Evaluate PS × PR . [4] (c) Find an equation of the line l which passes through the point Q and is parallel to the vector PR . (d) (e) [3] Find an equation of the plane  which contains the line l and passes through the point S. [4] Find the shortest distance between the point P and the plane  . [4] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 20 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 21 [MAA 3.17-3.19] PLANES 29. [Maximum mark: 25] [without GDC] Consider the points A(1, 2, 1), B(0, –1, 2), C(1, 0, 2) and D(2, –1, –6). (a) Find the vectors AB and BC . [2] (b) Calculate AB BC . [2] (c) Hence, or otherwise find the area of triangle ABC . [3] (d) Find the Cartesian equation of the plane P containing the points A , B and C . [3] (e) Find a set of parametric equations for the line L through the point D and perpendicular to the plane P . [3] (f) Find the point of intersection E , of the line L and the plane P . [4] (g) Find the distance from the point D to the plane P . [2] (h) Find a unit vector that is perpendicular to the plane P . [2] (i) The point F is a reflection of D in the plane P . Find the coordinates of F. 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Page 22 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 23 [MAA 3.17-3.19] PLANES 30. [Maximum mark: 16] [without GDC] The points A, B, C, D have the following coordinates A : (1, 3, 1) B : (1, 2, 4) C : (2, 3, 6) D : (5, – 2, 1). (a) (i) Evaluate the vector product AB × AC , giving your answer in terms of the unit vectors i, j, k. (ii) Find the area of the triangle ABC. [6] The plane containing the points A, B, C is denoted by Π and the line passing through D perpendicular to Π is denoted by L. The point of intersection of L and Π is denoted by P. (b) (i) Find the cartesian equation of Π . (ii) Find the cartesian equations of L. [5] (c) Determine the coordinates of P. [3] (d) Find the perpendicular distance of D from Π . 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Page 24 [MAA 3.17-3.19] PLANES 31. [Maximum mark: 16] A line l1 has equation [without GDC] x  2 y z 9   . Let M be a point on l1 with parameter μ. 2 3 1 (a) Express the coordinates of M in terms of μ. (b) The line l2 is parallel to l1 and passes through P(4, 0, –3). (i) (c) (ii) Express PM in terms of μ. Find the value of μ. (ii) Find the distance between l1 and l2. [5] The plane  1 contains l1 and l2. Find an equation for  1 , giving your answer in the form Ax + By + Cz = D. (e) [4] The vector PM is perpendicular to l1. (i) (d) Write down an equation for l2. [1] [4] The plane  2 has equation x  5 y  z  11 . Verify that l1 is the line of intersection of the planes  1 and  2 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 25 [2] [MAA 3.17-3.19] PLANES 32. [Maximum mark: 29] (a) Show that lines [without GDC] x2 y 2 z 3 x2 y 3 z 4   and   intersect and find 1 3 1 1 4 2 the coordinates of P, the point of intersection. [8] (b) Find the Cartesian equation of the plane Π that contains the two lines. [6] (c) The point Q (3, 4, 3) lies on Π . The line L passes through the midpoint of [PQ]. Point S is on L such that PS  QS  3 , and the triangle PQS is normal to the plane Π . Given that there are two possible positions for S, find their coordinates. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 26 [15] [MAA 3.17-3.19] PLANES 33. [Maximum mark: 12] (a) [with / without GDC]  2   2  1        The plane  1 has equation r =  1     1      3  . 1  8    9        2   1   1       The plane  2 has the equation r =  0   s 2   t 1 .  1   1   1       (b) (c) (i) For points which lie in  1 and  2 , show that,    . (ii) Hence, find a vector equation of the line of intersection of  1 and  2 . [5] y The plane  3 contains the line 2  x   z  1 and is perpendicular to 3 4 3i – 2j + k. Find the Cartesian equation of  3 . [4] Find the intersection of  1 ,  2 and  3 . [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 27 [MAA 3.17-3.19] PLANES 34. [Maximum mark: 14] (a) [without GDC] The line l1 passes through the point A(0, 1, 2) and is perpendicular to the plane x  4 y  3 z  0 . Find the Cartesian equation of l1 . (b) The line l2 is parallel to l1 and passes through the point P(3,  8,  11) . Find the vector equation of the line l2 . (c) [2] (i) [2] The point Q is on the line l1 such that PQ is perpendicular to l1 and l2 . Find the coordinates of Q . (ii) Hence find the distance between l1 and l2 . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 28 [10] [MAA 3.17-3.19] PLANES 35. [Maximum mark: 21] [with / without GDC] Consider the points A(2, –1, 0), B(3, 0, 1) and C(1, m, 2), where m  ℤ, m  0 . (a) (i) 2 , show that m  1 . 3 Determine the Cartesian equation of the plane ABC . Find the area of the triangle ABC . (i) The line L is perpendicular to plane ABC and passes through A . Find a vector equation of L . (ii) The point D(6, –7, 2) lies on L . Find the volume of the pyramid ABCD . (ii) (b) (c) (d) Find the scalar product BA  BC . ˆ  arccos Hence, given that ABC .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 29 [6] [4] [3] [8] [MAA 3.17-3.19] PLANES 36. [Maximum mark: 23] [without GDC] Two planes  1 and  2 are represented be the equations 3  2   2       1 : r   1     2     1  5  3 0        2 : 2x  y  2z  4 (a) (i)  2   2      Find  2    1   3  0     (ii) Show that the equation of  1 can be written as x  2 y  2 z  11 . (b) Show that  1 is perpendicular to  2 . (c) The line l1 is the line of intersection of  1 and  2 . Find the vector equation of l1 , giving the answer in parametric form. (d) [4] [5] The line l2 is parallel to both  1 and  2 , and passes through P(3,  5,  1) . Find an equation for l2 in Cartesian form. (e) [4] [3] Let Q be the foot of the perpendicular from P to the plane  2 . (i) Find the coordinates of Q . (ii) Find PQ . [7] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 30 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 31 [MAA 3.17-3.19] PLANES 37. [Maximum mark: 27] [with GDC] Consider the vectors a = i – j + k, b = i + 2 j + 4k and c = 2i –5 j – k. (a) Given that c = ma + nb where m, n  ℤ , find the value of m and of n . [5] (b) Find a unit vector, u, normal to both a and b. [5] (c) The plane  1 contains the point (1,  1,1) and is normal to b. The plane intersects the x , y and z axes at the points L , M and N respectively. (d) (e) (i) Find a Cartesian equation of  1 . (ii) Write down the coordinates of L , M and N . [5] The line through the origin, O , normal to  1 meets  1 at the point P . (i) Find the coordinates of P . (ii) Hence find the distance of  1 from the origin. [7] The plane  2 has equation x  2 y  4 z  4 . Calculate the angle between  2 and a line parallel to a. [5] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 32 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 33 [MAA 3.17-3.19] PLANES 38. [Maximum mark: 18] (a) [with / without GDC] Write the vector equations of the following lines in parametric form.  3 2     r1 =  2   m  1 7 2     1  4      r2 =  4   n  1  2  1      [2] (b) Hence show that these two lines intersect and find the point of intersection, A. [5] (c) Find the Cartesian equation of the plane Π that contains these two lines. [4] (d)   8  3     Let B be the point of intersection of the plane Π and the line r =   3     8  .  0   2     Find the coordinates of B. (e) [4] If C is the mid-point of AB, find the vector equation of the line perpendicular to the plane Π and passing through C. [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 34 [MAA 3.17-3.19] PLANES .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Page 35 [MAA 3.17-3.19] PLANES 39. [Maximum mark: 20] [without GDC] The points A, B, C have position vectors i + j + 2k, i + 2 j + 3k, 3i + k respectively and lie in the plane  . (a) Find the area of the triangle ABC; [6] (b) Hence, or otherwise, find the shortest distance from C to the line AB; [4] (c) Find the Cartesian equation of the plane  . [4] The line L passes through the origin and is normal to the plane  , it intersects  at D. (d) Find the coordinates of the point D; [5] (e) Find the distance of  from the origin. [1] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 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