The resultant of two vectors A and B is C . If the magnitude of B...

According to the question, $A+B=C \ldots . .(0)$ Also, when $B$ is doubled, $C$ becomes perpendicular to $A$. Let the new magnitude of $C$ be $C^{\prime}$ That means $A . C^{\prime}=0\left(A C^{\prime} \cos 90=0\right)>(A)(A+2 B)=0 \ldots \ldots .(1)>\left(A^{\wedge} 2+2 A B\right)=0>|A|=-2 A B$ To find the magnitude of $C$, square eq. (0) $>(A+B)^{\wedge} 2=|C|>|A|+|B|+2 A B$ but $2 A B=-|A| .$ from $(1)$ which means $|C|=|B|$