Vector Algebra · Question 37

Let a = i + j + k, b = 2 i + 2 j + k and d = a b. If c is a vector such that a c = | c|, | c - 2 a|2 = 8 and the angle between d and c is 4, then |10 - 3 b c| + | d c|2 is equal to .

Answer

6

Solution

d = a b = vmatrix i & j & k \ 1 & 1 & 1 \ 2 & 2 & 1 vmatrix = - i + j, | d| = 2

| c - 2 a|2 = | c|2 - 4 a c + 4| a|2

8 = | c|2 - 4| c| + 12

(| c| - 2)2 = 0 | c| = 2

d c = | d|| c| 4 = 2 2 1 2 = 2

| d c|2 = | d|2| c|2 - ( d c)2 = 8 - 4 = 4

Let c = x i + y j + z k:

x + y + z = 2, -x + y = 2, x2 + y2 + z2 = 4

y = x + 2, z = -2x

x2 + (x + 2)2 + 4x2 = 4 6x2 + 4x = 0 x = 0 or x = - 23

b c = 2x + 2y + z = 2x + 4

x = 0: b c = 4 |10 - 12| = 2

x = - 23: b c = 83 |10 - 8| = 2

|10 - 3 b c| + | d c|2 = 2 + 4 = 6

Answer: 6