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...

You may write a vector as either (a, b, c) or a~i + b~j + c ~k, but be careful: you will lose marks if you mis-write these notations.


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) or 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 =  −2 0 00 5 0 0 0 1 . [4 marks for each case] [Total: 12 marks] Question 6. Find the inverse of the matrix D =  1 3 00 1 1 2 0 0 . Show your working. [24 marks] Total: 100 marks 3