Let a = 2 i + j - 2 k and b = i - 4 j - k . Let c be a vector such that | c | = 3 and a c = b . If the angle between a and c is acute, then the value of | c - a |^2 is equal to :

Step-by-step solution

We are given a = 2 i + j - 2 k and b = i - 4 j - k . First, we calculate the squared magnitudes of a and b : | a |^2 = (2)^2 + (1)^2 + (-2)^2 = 4 + 1 + 4 = 9 | b |^2 = (1)^2 + (-4)^2 + (-1)^2 = 1 + 16 + 1 = 18 From the given relation a c = b , we take the squared magnitude on both sides: | a c |^2 = | b |^2 = 18 Using Lagrange's identity, we know that: | a c |^2 + ( a c )^2 = | a |^2 | c |^2 Substitute the known values into the identity: 18 + ( a c )^2 = (9)( 3 )^2 18 + ( a c )^2 = 27 ( a c )^2 = 9 Since the angle