Vectors a i + b j + k and 2 i - 3 j + 4 k are perpendicular to each other when 3a + 2b = 7 , the ratio of a to b is x/2 . The value of x is

Step-by-step solution

Let the given vectors be p = a i + b j + k and q = 2 i - 3 j + 4 k . Since the vectors are perpendicular, their dot product is zero: p q = 0 2a - 3b + 4 = 0 2a - 3b = -4 We are given another equation: 3a + 2b = 7 Solving the two equations simultaneously, multiply the first by 2 and the second by 3 : 4a - 6b = -8 9a + 6b = 21 Adding both equations: 13a = 13 a = 1 Substituting a = 1 in 3a + 2b = 7 : 3(1) + 2b = 7 2b = 4 b = 2 The ratio of a to b is a b = 1 2 . Given that the ratio is x 2 , we have: x 2 = 1 2 x = 1 An