If \(\vec F=2\hat i+\hat j-\hat k\) and \(\vec r=3\hat i+2\hat j-2\hat k,\) then the scalar and vector products of \(\vec F\) and \(\vec r\) have the magnitudes, respectively, as:
1. \(5, ~\sqrt3\)
2. \(4,~ \sqrt5\)
3. \(10, ~\sqrt2\)
4. \(10,~2\)

The vector sum of two forces is perpendicular to their vector difference. In this case, the two forces: 

1. Are equal

2. Have the same magnitude

3. Are not equal in magnitude

4. Cannot be predicted

If  $|$ $\overrightarrow{A}$ $+$ $\overrightarrow{B}$ $|$ $=$ $\left|\overrightarrow{A}\right|$ $$ $=$ $|$ $\overrightarrow{B}|$ then angle between A and B will be:

1.  $90°$

2. $120°$

3. $0°$

4. $60°$

$\begin{array}{l}\text{\hspace{0.17em}If\hspace{0.17em}}|\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}|=\sqrt{3}\overrightarrow{\mathrm{A}}\cdot \overrightarrow{\mathrm{B}}\text{, then the value of\hspace{0.17em}}\\ |\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|\text{\hspace{0.17em}is:\hspace{0.17em}}\end{array}$

1. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\frac{\mathrm{AB}}{\sqrt{3}})}^{1/2}$

2. A+B

3. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\sqrt{3}\mathrm{AB})}^{1/2}$

4. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\mathrm{AB})}^{1/2}$

If the angle between the vector $\overrightarrow{\mathrm{A}}\text{\hspace{0.17em}and\hspace{0.17em}}\overrightarrow{\mathrm{B}}$ is \(\theta\), the value of the product $(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{A}})\cdot \overrightarrow{\mathrm{A}}$ is equal to:

1. zero

2. BA²\(\sin\theta \cos \theta\)

3. BA²\(\cos\theta\)

4. BA²\(\sin\theta\)

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. \(45^\circ\)

$\stackrel{}{\mathrm{}}$\(\overrightarrow{A}\) 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:

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.  $\sqrt{3}$ $\mathrm{N}$