Let a and b be unit vectors such that the angle between them is 3 . Let c be a vector satisfying ( a + 2 b ) c = c (2 a - b ) . If c a = 7 , then | c |^2 is equal to :

Step-by-step solution

Given ( a + 2 b ) c = c (2 a - b ) Using the anti-commutative property of the cross product, we have: ( a + 2 b ) c = -(2 a - b ) c Bringing all terms to one side: ( a + 2 b ) c + (2 a - b ) c = 0 (3 a + b ) c = 0 This implies that c is collinear with (3 a + b ) , so: c = (3 a + b ) Taking the dot product with a : c a = (3 a a + b a ) Since a and b are unit vectors with an angle of 3 between them: a a = | a |^2 = 1 b a = | b || a | ( 3 ) = 1 2 Substitute these values into the dot product equation: 7 = (3(1) + 1 2 )