Let a , b , and c be three unit vectors such that the angle between any two of them is 3 . A vector u of magnitude 3 makes equal acute angles with a , b , and c . The value of | u ( a b + b c + c a )|^2 is equal to

Step-by-step solution

Since u makes equal angles with a , b , and c , by symmetry it is collinear with their sum. Let u = ( a + b + c ) . First, we find the magnitude of a + b + c : | a + b + c |^2 = | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) | a + b + c |^2 = 1 + 1 + 1 + 2 ( 1 2 + 1 2 + 1 2 ) = 6 | a + b + c | = 6 . Given | u | = 3 , we have | | 6 = 3 | | = 1 2 . Since u makes acute angles with the vectors, u a > 0 , which implies > 0 . Thus, = 1 2 . u = 1 2 ( a + b + c ) . We need to evaluate E = u ( a b + b c + c a ) . Substi