If vectors $\overrightarrow{{A}}=\cos \omega t \hat{{i}}+\sin \omega t \hat{j}$ and $\overrightarrow{{B}}=\cos \left(\frac{\omega t}{2}\right)\hat{{i}}+\sin \left(\frac{\omega t}{2}\right) \hat{j}$ are functions of time. Then, at what value of $t$ are they orthogonal to one another?
1. $t = \frac{\pi}{4\omega}$
2. $t = \frac{\pi}{2\omega}$
3. $t = \frac{\pi}{\omega}$
4. $t = 0$

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

1. $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{f}}$

2. $\overrightarrow{\mathrm{d}}+\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{f}}$

3. $\overrightarrow{\mathrm{d}}+\overrightarrow{\mathrm{e}}=\overrightarrow{\mathrm{f}}$

4. $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{e}}=\overrightarrow{\mathrm{f}}$

Three forces acting on a body are shown in the figure. To have the resultant force only along the y-direction, the magnitude of the minimum additional force needed is:
                 

1.  $0.5$ $\mathrm{N}$

2.  $1.5$ $\mathrm{N}$

3.  $\frac{\sqrt{3}}{4}$ $\mathrm{N}$

4.  $\sqrt{3}$ $\mathrm{N}$

$\overrightarrow{A}$ and $\overrightarrow B$ are two vectors and $\theta$ is the angle between them. If $\left|\overrightarrow A\times \overrightarrow B\right|= \sqrt{3}\left(\overrightarrow A\cdot \overrightarrow B\right),$ then the value of $\theta$ will be:

The vectors $\overrightarrow{\mathrm{A}}$ $\mathrm{and}$ $\overrightarrow{\mathrm{B}}$ are such that: $\left|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\right|=\left|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}\right|$.
The angle between the two vectors is:
1. $90^\circ$
2. $60^\circ$
3. $75^\circ$
4. $45^\circ$