1 Two lines have equations
1
 1 
  1
 3 
 
 
 
 
r   5   s   2  and r    1  t  4  .
 2
 3 
 10 
  5
 
 
 
 
(i)
Show that the lines meet, and find the point of intersection.
(ii)
Calculate the acute angle between the lines.
2. The points A and B have position vectors 4i –2j –3k and i + j + 3k respectively, O is the
origin.
(i)
Find, in vector form, an equation for the line L that passes through A and B
(ii)
Find the perpendicular distance from O to line L.
(iii) Show that the line through O parallel to the vector i + 2j +3k does not intersect L .
3. Two planes P1 and P2 have equations
5x +y –2z = 0 and 2x +3y +z =1.
(i)
find the angle between the two planes
(ii)
Find the equation of the line of intersection of P1 and P2 ,giving your answer in the
form r = a + tb.
 5
1
 
 
4 The line L has equation r   8   t  0  and the plane P has equation 2x –2y –z –5 = 0.
1
8
 
 
(i)
Find the position vector of the point at which L and P intersect.
(ii)
Find also the acute angle between L and P, giving your answer correct to the nearest
degree.
 3 
 2
 
 
5. The plane P has equation (r-a).n = 0 ,where a   1  and n =  1  . The point B has
  2
1
 
 
 4
 
position vector  1  .Find the perpendicular distance from B to P.
 2
 
6. The planes P1 and P2 have equations x + y – z = 0 and 2x – 4y + z +12 = 0. respectively.
 p
 
(i)
Find numbers p,q and r such that the vector  q  is parallel to both P1 and P2.
r
 
(ii)
Hence find the equation of the plane through the point (3,8,2) which is perpendicular
to both P1 and P2.
7.The point A has coordinates (3,-1,5) and the line L has equation
 8 
  6
 


r  0  s 1 
  1
 4 
 


Find the coordinates of the point B on l such that AB is perpendicular to L.
The plane  has equation
 1 
 
r    1  15 .
 3 
 
Find the coordinates of the point C where L intersects 
Find a vector perpendicular to the plane ABC. Hence show that the acute angle between 
and the plane ABC is 68,correct to the nearest degree.
8. The plane 1 and 2 have equations
 3
 
3x - 4y +12z =3 and r   0   1 .respectively.
 2
 
1
 0
 
 
The line L has equation r   7   s  8 
 2
 2
 
 
 3 
 
And the point A has position vector   6 
 4 
 
(i)
Calculate to the nearest one tenth of a degree, the acute angle between planes
1 and 2 .
(ii)
(iii)
(iv)
Calculate the shortest distance from A to the plane 1
Find the position vectors of the two points in which the line L meets the planes
1 and 2
Find an equation of the plane containing the line L and the point A.
9. Find the perpendicular distance from the point (2,3,4) to the plane 4x-3y +z =7.
Ans: 326/26
10. The lines l and m have vector equations
 5
 4
 
 
r1   1      1
 2
 1
 
 
(i)
(ii)
(iii)
and
1
  2
 


r2   2     5 
1
 7 
 


respectively
Show that l and m intersect, find the position of their point of intersection.
Find the acute angle between the two lines, giving your answer to the nearest
degree.
Find the equation of the plane containing l and m, giving
your answer in the form ax + by + cz = d.
11.Two planes have vector equations
r.(2i -3j-k) =14
and
r.(11i +j-2k) = 42
(a) Find the acute angle between these two planes
(b) Determine a vector equation of the line of intersection of these two planes.
Ans: [ [60] r =.(2i 20k) +(i j + 5k) ]
12. For some value of the scalar constant m, the lines with equations
r = (2 + )i + (7+)j + (m - 7)k
and
r = (3)i + (11+3)j + (115)k
meet at the point P. Determine the value of m and the position vector of P.
(b) Find in the form r.n = d , the equation of the plane which contains the two lines.
(c) Find the angle between the line r =2i j +3k + t(i +j +4k)
and the plane r.(2i j 3k) = 0
(d) Find the coordinates of the point where the line r =2i +j +3k + s(i j 4k)
Meets the plane r.(2i +3j k) = 34
Ans :[m = 22.4 P = 4.1i +7.7j +17.5k r.(3i +26j +5k) = 34 ,43.9,(12,-9,-37).]
13.(a) Show that the line r = 5i +j +2k + s(2i +j k) is parallel to the plane 2x 3y + z =10.
(b) Find a vector equation of for the line of intersection of the planes
5x +2y +z = 8.
x +y +z = 4.
What is the angle between the two planes?
[r = 4j + t(i 4j +3k) ; 32.5,