The resultant $\vec{R}$ of $\vec{P}$ and $\vec{Q}$ is perpendicular to $\vec{P}$. Also,$|\vec{P}| = |\vec{R}|$. Find the angle between $\vec{P}$ and $\vec{Q}$.

Two vectors $\vec A$ and $\vec B$ have magnitudes $2$ and $1$ respectively. If the angle between $\vec A$ and $\vec B$ is $60^{\circ}$,then which of the following vectors may be equal to $\frac{\vec A}{2} - \vec B$?

The figure shows three vectors $\vec{a}, \vec{b}$ and $\vec{c}$,where $R$ is the midpoint of $PQ$. Which of the following relations is correct?

Statement $I:$ Two forces $(\overrightarrow{P}+\overrightarrow{Q})$ and $(\overrightarrow{P}-\overrightarrow{Q})$,where $\overrightarrow{P} \perp \overrightarrow{Q}$,act at an angle $\theta_{1}$ to each other,and the magnitude of their resultant is $\sqrt{3(P^{2}+Q^{2})}$. When they act at an angle $\theta_{2}$,the magnitude of their resultant becomes $\sqrt{2(P^{2}+Q^{2})}$. This is possible only when $\theta_{1} < \theta_{2}$.
Statement $II:$ In the situation given above,$\theta_{1} = 60^{\circ}$ and $\theta_{2} = 90^{\circ}$.
In the light of the above statements,choose the most appropriate answer from the options given below.

There are two force vectors,one of $5\, N$ and the other of $12\, N$. At what angle should the two vectors be added to get a resultant vector of $17\, N$,$7\, N$,and $13\, N$ respectively?

When $n$ vectors of different magnitudes are added,we get a null vector. Then the value of $n$ cannot be