A particle moves from a point \(\left(\right. - 2 \hat{i} + 5 \hat{j} \left.\right)\) to \(\left(\right. 4 \hat{j} + 3 \hat{k} \left.\right)\) when a force of \(\left(\right. 4 \hat{i} + 3 \hat{j} \left.\right)\) \(\text{N}\) is applied. How much work has been done by the force?

If vectors $\overrightarrow{\mathrm{A}}=\mathrm{cos\omega t}\hat{\mathrm{i}}+\mathrm{sin\omega t}\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}}=\cos$ $\frac{\mathrm{\omega t}}{2}\hat{\mathrm{i}}+\sin$ $\frac{\mathrm{\omega t}}{2}\hat{\mathrm{j}}$ are functions of time. Then, at what value of t are they orthogonal to one another?

1. $\mathrm{t}=\frac{\mathrm{\pi }}{4\mathrm{\omega }}$

2. $\mathrm{t}=\frac{\mathrm{\pi }}{2\mathrm{\omega }}$

3. $\mathrm{t}=\frac{\mathrm{\pi }}{\mathrm{\omega }}$

4. $\mathrm{t}=0$

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.  $\sqrt{3}$ $\mathrm{N}$

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

1. 60^o

2. 45^o

3. 30^o

4. 90^o

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. \(45^\circ\)

If a vector ($2\hat{\mathrm{i}}+3\hat{j}+8\hat{k}$) is perpendicular to the vector ($4\hat{\mathrm{i}}-4\hat{j}+\mathrm{\alpha }\hat{k}$), then the value of $\mathrm{\alpha \; is:}$

1. -1

2. $-\frac{1}{2}$

3. $\frac{1}{2}$

4. 1

If the angle between the vector $\overrightarrow{\mathrm{A}}\text{ and }\overrightarrow{\mathrm{B}}$ is θ, the value of the product $(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{A}})\cdot \overrightarrow{\mathrm{A}}$ is equal to:

1. zero

2. BA²sinθcosθ

3. BA²cosθ

4. BA²sinθ