If the lengths of three vectors a , ~b and c are 5,12,13 units respectively, and each one is perpendicular to the sum of the other two, then | a + b + c |=

Step-by-step solution

Given vectors a , b , and c with magnitudes | a | = 5 , | b | = 12 , and | c | = 13 , where each vector is perpendicular to the sum of the other two. This implies the dot products: a ( b + c ) = 0 , b ( a + c ) = 0 , and c ( a + b ) = 0 . Expanding and summing these three equations yields: 2( a b + b c + c a ) = 0 , so a b + b c + c a = 0 . The magnitude squared of the resultant vector R = a + b + c is: | R |^2 = | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) . Substituting the known values gives: | R |^2 = 5^2