If | a | = 3 , | b | = 4 , | c | = 5 such that each vector is perpendicular to the sum of the other two, then | a + b + c | is equal to

Step-by-step solution

Given that each vector is perpendicular to the sum of the other two, we have: a ( b + c ) = 0 a b + a c = 0 b ( c + a ) = 0 b c + b a = 0 c ( a + b ) = 0 c a + c b = 0 Adding the above three equations, we get: 2( a b + b c + c a ) = 0 Using the identity for the square of the sum of three vectors: | a + b + c |^2 = | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) Substituting the given magnitudes | a | = 3 , | b | = 4 , and | c | = 5 : | a + b + c |^2 = 3^2 + 4^2 + 5^2 + 0 | a + b + c |^2 = 9 + 16 + 25 = 50 | a +