Two forces are such that the sum of their magnitudes is $18 \,N$ and their resultant is perpendicular to the smaller force and the magnitude of the resultant is $12 \,N$. Then the magnitudes of the forces are:

Two vectors $\vec A$ and $\vec B$ have equal magnitudes. The magnitude of $(\vec A + \vec B)$ is $n$ times the magnitude of $(\vec A - \vec B)$. The angle between $\vec A$ and $\vec B$ is

What is the angle between $\overrightarrow P$ and the resultant of $(\overrightarrow P + \overrightarrow Q)$ and $(\overrightarrow P - \overrightarrow Q)$?

$A$ vector of length $\ell$ is turned by an angle $\theta$. Find the change in the position vector of the tip.

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. Then the vector $\vec{A} + \vec{B}$ is perpendicular to:

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ have equal magnitudes. If the magnitude of $\overrightarrow{A} + \overrightarrow{B}$ is equal to two times the magnitude of $\overrightarrow{A} - \overrightarrow{B}$,then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ will be .......................