Let a , ~b and c be vectors of magnitude 2,3 and 4 respectively. If a is perpendicular to ( b + c ), b is perpendicular to ( c + a ) and c is perpendicular to ( a + b ), then the magnitude of a + b + c is

Step-by-step solution

Given vector magnitudes : | a | = 2 , | b | = 3 , | c | = 4 . The perpendicularity conditions yield the dot product equations: a b + a c = 0 b c + b a = 0 c a + c b = 0 Let x = a b , y = b c , z = c a . The system becomes: x + z = 0 y + x = 0 z + y = 0 From the first equation, z = -x . From the second, y = -x . Substituting into the third gives z + y = -x + (-x) = -2x = 0 , so x = 0 . Hence y = 0 and z = 0 , confirming the vectors are mutually orthogonal. The magnitude of the sum is found from: | a + b + c |^2 = |