The vectors a , ~b and c are such that | a |=2,| ~b |=4,| c |=4 . If the projection of b on a is equal to projection of c on a and b is perpendicular to c , then the value of | a + b - c | is

Step-by-step solution

Given vectors a , b , and c with magnitudes | a | = 2 , | b | = 4 , and | c | = 4 . The projection of b on a equals that of c on a , so b a = c a , which implies a ( b - c ) = 0 . Also, b c , so b c = 0 . To compute | a + b - c | , square and expand: | a + b - c |^2 = | a |^2 + | b - c |^2 + 2 a ( b - c ) . Since a ( b - c ) = 0 , this simplifies to | a |^2 + | b - c |^2 . Now, | b - c |^2 = | b |^2 + | c |^2 - 2 b c . Given b c = 0 , it reduces to | b |^2 + | c |^2 = 16 + 16 = 32 . Substitute to get | a + b - c |^