Question 1.
(a) Two particles bounce off each other and continue moving in different
directions. The first particle’s direction is defined by ~v = 6~i−~j+~k and
the second particle’s direction is defined by ~w = 2~i + 2~j − ~k. Use the
vector product to find the angle between the paths of the two particles.
Show your working. [9 marks]
(b) Four points A(2, 0, 1), B(−1, 2, 3), C(3, 2, 2) and D(3,−6,−3) are cho-
sen to be the vertices of a 3-dimensional shape. Show that the con-
struction of such a 3-dimensional shape is impossible in this case. [9 marks]
(c) Determine whether the following two planes x + 4y − z = 7 and 5x−
3y− 7z = 11 are parallel, orthogonal, coincident (that is, the same) o
none of these. [6 marks]
[Total: 24
marks]
Question 2.
Let ~u =~i−~j + 2~k, ~v = 2~i− 3~j + ~k.
(a) Calculate the dot product of ~u and ~v. Show your working. [6 marks]
(b) Calculate the vector product of ~u and ~v. Show your working. [6 marks]
[Total: 12
marks]
Question 3.
Given matrix A =
1 2 52 −1 3
1 1 −1
(a) Write the matrix 4A; [4 marks]
(b) Use matrix multiplication to find A2 = A ·A. Show your working; [6 marks]
(c) Find detA by expanding any column. Show your working. [6 marks]
[Total: 16
marks]
Question 4.
Suppose matrix product AB is defined.
(a) If A is 3 × 6 and B is a column matrix, give the dimensions of B and
AB. [4 marks]
(b) If A is the identity matrix and B is 6 × 6, what size is A? [4 marks]
(c) If A is 3 × 8 and AB is 3 × 7, what size is B? [4 marks]
[Total: 12
marks]
2
Question 5.
Find the determinant of the matrices below by inspection. Give your reason
in each case.
B =
1 7 −4 −5
1 5 7 −4
0 21 −12 −15
5 25 35 −20
C =
−1 2 −30 5 7
0 0 9
D =
− XXXXXXXXXX
0 0 1
.
[4 marks fo
each case]
[Total: 12
marks]
Question 6.
Find the inverse of the matrix D =
XXXXXXXXXX
2 0 0
. Show your working. [24 marks]
Total: 100 marks
3