Q.The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to vector $\vec{A}$, and its magnitude is equal to half the magnitude of vector $\vec{B}$. Then the angle between $\vec{A}$ and $\vec{B}$ is

Two vectors cannot be added by ordinary arithmetic because direction matters. The triangle law gives the correct geometric recipe: draw $\vec A$ and $\vec B$ as two sides of a triangle taken in the same order — draw $\vec A$ first, then draw $\vec B$ starting from the head (tip) of $\vec A$; the resultant $\vec R=\vec A+\vec B$ is then the third side, drawn directly from the tail of $\vec A$ to the head of $\vec B$. Its magnitude and direction (angle $\alpha$ from $\vec A$) are found by dropping a perpendicular in the resulting triangle, giving, for an angle $\theta$ between $\vec A$ and $\vec B$:

$R=\sqrt{A^2+B^2+2AB\cos\theta}, \qquad \tan\alpha=\frac{B\sin\theta}{A+B\cos\theta}$ …