Which of the four arrangements in the figure correctly shows the vector addition of two forces $\overrightarrow {{F_1}} $ and $\overrightarrow {{F_2}} $ to yield the third force $\overrightarrow {{F_3}} $?

The vector that must be added to the vectors $\hat{i} - 3\hat{j} + 2\hat{k}$ and $3\hat{i} + 6\hat{j} - 7\hat{k}$ so that the resultant vector is a unit vector along the $y$-axis is

If $\overrightarrow{A} = 3\widehat{i} + 2\widehat{j}$ and $\overrightarrow{B} = \widehat{i} + \widehat{j} - 2\widehat{k}$,find their sum using the algebraic method.

Magnitudes of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are $4$ units and $3$ units respectively. If these vectors are $(i)$ in the same direction $(\theta = 0^{\circ})$ and $(ii)$ in the opposite direction $(\theta = 180^{\circ})$,find the magnitude of the resultant vector in each case.

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A} = a \hat{i}$ and $\vec{B} = a(\cos \omega t \hat{i} + \sin \omega t \hat{j})$,where $a$ is a constant and $\omega = \pi / 6 \text{ rad s}^{-1}$. If $|\vec{A} + \vec{B}| = \sqrt{3}|\vec{A} - \vec{B}|$ at time $t = \tau$ for the first time,the value of $\tau$,in seconds,is . . . . . .

If $|{\overrightarrow V _1} + {\overrightarrow V _2}| = |{\overrightarrow V _1} - {\overrightarrow V _2}|$ and ${V_2}$ is finite,then