Given vectors a = [1, -3], b = [2, 3], and c = [8, 3], find m and n such that c = ma + nb. Then find |d| if d = c - 2a.

Here's the solution to question 12. 12. (a) (i) Find m and n such that c = ma + nb Step 1: Substitute the given vectors into the equation. 8 \\ 3 = m 1 \\ -3 + n 2 \\ 3 8 \\ 3 = m \\ -3m + 2n \\ 3n 8 \\ 3 = m + 2n \\ -3m + 3n Step 2: Form a system of linear equations from the components. m + 2n = 8 (1) -3m + 3n = 3 (2) Step 3: Solve the system of equations. From equation (2), divide by 3: -m + n = 1 n = m + 1 Substitute n = m + 1 into equation (1): m + 2(m + 1) = 8 m + 2m + 2 = 8 3m = 6 m = 2 Substitute m = 2 back into n = m + 1: n = 2 + 1 n = 3 The values are m=2 and n=3. m=2, n=3 12. (a) (ii) Find |d|, if d = c - 2a Step 1: Calculate vector d. d = 8 \\ 3 - 2 1 \\ -3 d = 8 \\ 3 - 2 \\ -6 d = 8 - 2 \\ 3 - (-6) d = 6 \\ 9 Step 2: Calculate the magnitude of d. |d| = sqrt(6^2 + 9^2) |d| = sqrt(36 + 81) |d| = sqrt(117) Step 3: Simplify the radical. |d| = sqrt(9 × 13) |d| = 3sqrt(13) The magnitude of d is 3sqrt(13). |d| = 3sqrt(13) 12. (b) If x+3 \\ 2 - y \\ x+y = 2 \\ 1 , find the values for x and y. Step 1: Perform the vector subtraction on the left side. (x+3) - y \\ 2 - (x+y) = 2 \\ 1 x+3-y \\ 2-x-y = 2 \\ 1 Step 2: Form a system of linear equations from the components. x + 3 - y = 2 x - y = -1 (1) 2 - x - y = 1 -x - y = -1 (2) Step 3: Solve the system of equations. Add equation (1) and equation (2): (x - y) + (-x - y) = -1 + (-1) -2y = -2 y = 1 Substitute y = 1 into equation (1): x - 1 = -1 x = 0 The values are x=0 and y=1. x=0, y=1 That's 2 down. 3 left today — send the next one.