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MCV 4U 1. Given a) y z Unit 2: Algebraic Vectors Extra Review Solutions x 2iˆ ˆj 5kˆ, y 1,2,2, z 3,2,4 A(1,3,4), B(3,2,7) Find: b) z x c) BA d) BA e) ŷ f) the angle between x and g) 2 x 3 y z h) 2 x y 3x 2 y i) 3x 2 y 2 x y z 2. Draw the position vector u 2,3,5 . Be sure to draw the x, y, and z axis and label them. 3. Using vectors demonstrate that the following points are collinear. A(-1,3,-7), B(-3,4,2) and C(5,0,-34) 4. A 60 N force is applied to the end of a 30 cm wrench and makes an angle of 40 with the handle of the wrench. What is the magnitude of the torque on the bolt at the other end of the wrench. 5. Find the work done if a force of 35 N in the direction of r 4,5,8 moves an object from G(1,7,-3) to H(5,3,-2). 44 41.6 41.6 150.4 J 6. A) If x 3t ,5,1 and y t 2, t 3,5 are perpendicular, find all value(s) for t. B) For x 14,4k ,7 and y 2c,8,21 , find values for k and c if x is parallel with y . 7. Given W(-1,4,2), X(6,-2,3) and Y(-3,5,1), find: a) area of triangle WXY b) the co-ordinates of the point Z if WXYZ is a parallelogram. 8. Given u 3,1,4 and v 2,4,1 , find the Pr oj u onto v as an algebraic vector. 9. If OA, OB, and OC are three edges of a parallelepiped where O is (0,0,0), A is (2,4,-3), B is (4,6,2), and C is (5,0,-2), find the coordinates of the other vertices of the parallelepiped and find the volume of the parallelepiped. V ppp'd a b c V ppp'd ( 26, 16, 4) (5, 0, 2 V ppp'd 130 8 V ppp'd 138 u 3 12. Given : and a 5 and b 8 Find: a) a b {29.73 ê in} b) the area of the parallelogram formed by a and b . {29.73 units squared} 13. Given vectors b (2,1,7) and c (5,2,1) . Find c b {(-15,-37,-1)} 14. Given A(2,2,5 ) , B(2, 15 , ) and C( 6 , 1, 5 ) , find the area of triangle ABC. {25.7 u squared} 15. Determine the magnitude of the moment of the force about the fixed point A for each of the following: a) b) {704.2 Joules} 16. Find a vector to both a = (1,1,2) and b = (2,1,- 2). {769.1 Joules} 17. Given: a (1,2,2) and b (2,1,3) find each of the following: a. the (vector) projection of a on b c. any vector to a d. a unit vector to both a and b . e. the area of a trianglewith adjacent sides a and b f. the angle between a and b 18. Given that d 5 , c 8 and the angle between d and c is 36 . Find 3d c 2d c { 117.6 ê } 19. Expand and simplify: a) 2 x y 3x 2 y b) 3x 2 y 2 x y