If $P + Q = R$ and $|P| = |Q| = \sqrt{3}$ and $|R| = 3$,then the angle between $P$ and $Q$ is

For two vectors $\vec{a}$ and $\vec{b}$,if $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$,then the angle between $\vec{a}$ and $\vec{b}$ is:

Six vectors,$\overrightarrow a$ through $\overrightarrow f$,have the magnitudes and directions indicated in the figure. Which of the following statements is true?

Four forces act on a stationary particle at the origin of a coordinate system: $\overrightarrow{F_1} = 3\hat{i} - \hat{j} + 9\hat{k}$,$\overrightarrow{F_2} = 2\hat{i} - 2\hat{j} + 16\hat{k}$,$\overrightarrow{F_3} = 9\hat{i} + \hat{j} + 18\hat{k}$,and $\overrightarrow{F_4} = \hat{i} + 2\hat{j} - 18\hat{k}$. In which plane will the particle move under the influence of these forces?

Maximum and minimum magnitudes of the resultant of two vectors of magnitudes $P$ and $Q$ are in the ratio $3:1.$ Which of the following relations is true?

$A$ vector $\vec Q$ which has a magnitude of $8$ is added to the vector $\vec P$ which lies along the $x$-axis. The resultant of the two vectors lies along the $y$-axis and has a magnitude twice that of $\vec P$. The magnitude of $\vec P$ is: