Let a , ~b , c be three vectors such that a + b + c = 0 ,| a |=3,| ~b |=4 , | c |=5 , then a ~b + b c + c a =

Step-by-step solution

Given that a + b + c = 0 and | a | = 3 , | b | = 4 , | c | = 5 , to find a b + b c + c a . The magnitude of the sum squared expands as | a + b + c |^2 = | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) . Since a + b + c = 0 , the left side is zero: 0 = 3^2 + 4^2 + 5^2 + 2( a b + b c + c a ) 0 = 9 + 16 + 25 + 2( a b + b c + c a ) 0 = 50 + 2( a b + b c + c a ) Solving gives a b + b c + c a = -25 Final answer: -25