Let a and b be two unit vectors such that the angle between them is 6 . If the vector c is given by c = 2 a + 4 b + 2 3 ( a b ) , then the value of | c |^2 - 4( c a ) is

Step-by-step solution

First, we calculate c a : c a = (2 a + 4 b + 2 3 ( a b )) a c a = 2| a |^2 + 4( a b ) + 0 c a = 2(1) + 4 ( 6 ) = 2 + 4 ( 3 2 ) = 2 + 2 3 . Next, we calculate | c |^2 . Since a b is perpendicular to both a and b , the vector 2 a + 4 b is orthogonal to 2 3 ( a b ) . Using the Pythagorean theorem for vectors: | c |^2 = |2 a + 4 b |^2 + |2 3 ( a b )|^2 Expanding the first term: |2 a + 4 b |^2 = 4| a |^2 + 16| b |^2 + 16( a b ) = 4(1) + 16(1) + 16 ( 6 ) = 20 + 16 ( 3 2 ) = 20 + 8 3 . Expanding the second term: |2 3 ( a