If a vector B is added to A, under what condition does the resultant vector have a magnitude equal to A+B?

Let the resultant vector of the two vectors $A$ and $B$ be $C$ then $C=A+B$.

From triangle law of vector addition, \[C=\sqrt{A^2+B^2+2AB\cos\theta\] where $\theta$ is the angle between $A$ and $B$.

\[\therefore A+B=\sqrt{A^2+B^2+2AB\cos\theta\] \[(A+B)^2=\sqrt{A^2+B^2+2AB\cos\theta\] \[2AB=2AB\cos\theta\] \[\cos\theta=1\] \[\therefore\theta=0°\]

Hence, the resultant vector have a magnitude to $A+B$ when the angle between $A$ and $B$ is $0°$.