Pre-U TOPIC REVIEW
Pre-U Term 2a
vectors
1. The position vectors of points C and D are given by OC  3i  4 j  2k and OD  2i  j  2k .
a) Write down, in surd form, the length of the vector OC.
(2 marks)
b) Hence write down a unit vector in the direction of OD.
(1 mark)
c) Write down the vector CD.
(1 mark)
d) Hence, calculate the distance between the points C and D, leaving your answer in surd
form.
(1 mark)
2. Write down the vector equation of the line given the following information :
a) passing through the point (4, 3) in the direction 2i + j.
(1 mark)
b) passing through the points (2, 5) and (4, −2).
(2 marks)
c) passing through the points with position vectors 2i + j − 6k and −4i + 3j − 3k.
(2 marks)
3. Find the point of intersection of the lines given by r  (i  2 j  k )   (3i  j  2k ) and
s  (6i  3j  k )   (i  4 j  3k ) .
(7 marks)
4. Show that the lines r  (i  2 j  3k )   (i  j  k ) and s  (3i  7 j  k )   (2i  j  3k ) are
skew.
(6 marks)
5. Use the scalar product to find the acute angle, in degrees, between the following lines:
a) r  (i  2 j  k )   (3i  j  2k ) and r  (4i  7 j  3k )   (2i  7 j  k ) .
(3 marks)
b) r  (2 j  5k )   (4i  2 j  3k ) and r  (i  6k )   (3i  6 j  5k ) .
(3 marks)
6. a) Find a vector equation of the straight line l through the points (2, 1, 4) and (3, 0, 2).
(2 marks)
b) The distinct points P and Q lie on l and are each a distance
Find the position vectors of P and Q.
17 units from the origin.
(5 marks)
7. The points A, B and C have coordinates (24, 6, 0), (30, 12, 12) and (18, 6, 36) respectively.
a) Find a vector equation for the line passing through the points A and B.
(1 mark)
b) The point P lies on the line passing through A and B. Show that CP can be expressed as
(6  t )i  tj  (2t  36)k , where t is a parameter.
(4 marks)
c) Given that CP is perpendicular to AB , find the coordinates of P.
(6 marks)
d) Hence, or otherwise, find the area of the triangle ABC, giving your answer to 3 significant
figures. (Hint : draw a diagram!)
(3 marks)
A– LEVEL TOPIC REVIEW : ANSWERS
unit C4
1. a)
b)
32  42  22  29
1
(3i  4 j  2k )
29
c) i  5 j  4k
vectors
M1A1
B1
B1
B1
2. a) r  (4i  3 j)   (2i  j) o.e.
A1
b) r  (2i  5 j)   (2i  7 j) o.e.
M1A1
c) r  (2i  j  6k )   (6i  2 j  3k ) o.e.
M1A1
6 2  14  21  17
y : 2    3  4 
M1
  13   2 13 ,  32 , 3 13 
A1
  2  (4,  1, 0)
A1
7. a) r  (24i  6 j)   (6i  6 j  12k ) o.e. B1
(24i  6 j)   (6i  6 j  12k )
B2
(6i  36k )   (6i  6 j  12k )
M1A1
(6i  36k )  t (i  j  2k )
check in remaining equation
M1A1
(6  t )i  tj  (2t  36)k
B1
4. x : 1    3  2 
y : 2  7
z : 3    1  3
B2
solve a pair of equations
M1A1
find inconsistency in last equation
M1A1
5. a) 6  7  2  14 54 cos 
  83.7
b) 12  12  15  29 70 cos 
M1A1
A1
M1A1
  101.5
180 101.5  78.5
A1
(3  1)(  2)  0
solve a pair →   3,   2.
(−8, 5, 5)
M1
3  7  2  0
b) CP  CO  OP
(18i  6 j  36k )
3. x : 1  3  6  
z :  1  2  1  3
b) (2   )2  (1   )2  (4  2 ) 2  17
2
12  52  42  42
d)
6. a) r  (2i  j  4k )   (i  j  2k ) o.e.
M1A1
c) CP AB  0
A1
A1
M1
M1
6(6  t )  6t  12(2t  36)  0
A1
36  6t  6t  24t  432  0
36t  396
t  11
A1
OP  OC  CP
(18i  6 j  36k )  (17i  11j  14k )
M1
(35, 17, 22)
B1
d) area   AB  CP
1
2
1
2
62  62  122 172  112  142
181 square units
B1
M1
A1
M1
A1
A1