Pure Mathematics Year 2 (A Level) Unit Test 12: Vectors
1 a The coordinates of A and B are (−1, 7, k) and (4, 1, 10) respectively. Given that
the distance from A to B is units, find the possible values of the constant k. (3 marks)
b For the larger value of k, find the unit vector in the direction of . (3 marks)
2 A triangle has vertices A(−2, 0, −4), B(−2, 4, −6) and C(3, 4, 4). By considering the
side lengths of the triangle, show that the triangle is a right-angled triangle. (6 marks)
3 Find the angle that the vector makes with the positive y-axis. (3 marks)
4 a Show that in with and ,
to one decimal place. (7 marks)
b Hence find and . (3 marks)
5 Given that , find the values of a, b
and c. (6 marks)
6 A particle of mass 3 kg is acted on by three forces, ,
and .
a Find the resultant force R acting on the particle. (2 marks)
b Find the acceleration of the particle, giving your answer in the form
(2 marks)
c Find the magnitude of the acceleration. (2 marks)
d Given that the particle starts at rest, find the exact distance travelled by the
particle in the first 10 s. (3 marks)
7 The diagram shows a cuboid whose vertices are O, A, B, C, D, E, F and G. a, b and
c are the vectors , and respectively. The points M and N lie on OA
such that . The points K and L lie on EF such that
Figure 1
Prove that the diagonals KN and ML bisect each other at P. (10 marks)
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1 a The coordinates of A and B are (−1, 7, k) and (4, 1, 10) respectively. Given that
the distance from A to B is units, find the possible values of the constant k. (3 marks)
b For the larger value of k, find the unit vector in the direction of . (3 marks)
2 A triangle has vertices A(−2, 0, −4), B(−2, 4, −6) and C(3, 4, 4). By considering the
side lengths of the triangle, show that the triangle is a right-angled triangle. (6 marks)
3 Find the angle that the vector makes with the positive y-axis. (3 marks)
4 a Show that in with and ,
to one decimal place. (7 marks)
b Hence find and . (3 marks)
5 Given that , find the values of a, b
and c. (6 marks)
6 A particle of mass 3 kg is acted on by three forces, ,
and .
a Find the resultant force R acting on the particle. (2 marks)
b Find the acceleration of the particle, giving your answer in the form
(2 marks)
c Find the magnitude of the acceleration. (2 marks)
d Given that the particle starts at rest, find the exact distance travelled by the
particle in the first 10 s. (3 marks)
7 The diagram shows a cuboid whose vertices are O, A, B, C, D, E, F and G. a, b and
c are the vectors , and respectively. The points M and N lie on OA
such that . The points K and L lie on EF such that
Figure 1
Prove that the diagonals KN and ML bisect each other at P. (10 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
