The resultant of two vectors A and B is C . If the magnitude of B is doubled, the new resultant vector becomes perpendicular to A , then the magnitude of C is

Step-by-step solution

Let C = A + B be the original resultant vector. When the magnitude of B is doubled, the new resultant vector becomes C ' = A + 2 B , which is perpendicular to A . For perpendicular vectors, C ' A = 0 . Expanding the dot product gives ( A + 2 B ) A = 0 , which simplifies to A^2 + 2( A B ) = 0 . Solving yields A B = - A^2 2 . The magnitude of C is found from | C |^2 = ( A + B ) ( A + B ) = A^2 + B^2 + 2( A B ) . Substituting the dot product gives | C |^2 = A^2 + B^2 - A^2 = B^2 . Thus, | C | = B .