If for two vectors \(\overrightarrow{A}\) and \(\overrightarrow {B}\), \(\overrightarrow {A}\times \overrightarrow {B}=0\), then the vectors:

$\overrightarrow{\mathrm{A}}$ 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:

The linear velocity of a rotating body is given by $\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{\omega }}\times \overrightarrow{\mathrm{r}}$, where $\mathrm{\omega }$ is the angular velocity and r is the radius vector. The angular velocity of a body, $\overrightarrow{\mathrm{\omega }}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and their radius vector is  $\overrightarrow{\mathrm{r}}=4\hat{\mathrm{j}}-3\hat{\mathrm{k}}, \mathrm{then\; value\; of} \left|\overrightarrow{\mathrm{v}}\right|$ will be:

1.  $\sqrt{29} \mathrm{units}$

2.  $31 \mathrm{units}$

3.  $\sqrt{37} \mathrm{units}$

4.  $\sqrt{41} \mathrm{units}$

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, then $\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right|=?$ 

If $\left|\overrightarrow{\mathrm{A}}\right| \ne \left|\overrightarrow{\mathrm{B}}\right|$ and $\left|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}\right|$ = $\left|\overrightarrow{\mathrm{A}} · \overrightarrow{\mathrm{B}}\right|$, then 

1.  $\overrightarrow{\mathrm{A}} \perp \overrightarrow{\mathrm{B}}$

2.  $\overrightarrow{\mathrm{A}} \parallel \overrightarrow{\mathrm{B}}$

3.  $\overrightarrow{\mathrm{A}}$ is antiparallel to $\overrightarrow{\mathrm{B}}$

4.  $\overrightarrow{\mathrm{A}}$ is inclined to $\overrightarrow{\mathrm{B}}$ at an angle of 45$°$

The scalar and vector product of two vectors, $\overrightarrow{\mathrm{a}} = \left(3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{b}} = \left(-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right)$ is equal to:

1. -25 & $\left(7\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$

2. 25 & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$

3. 0 & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right)$

4. -25 & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$

Which of the following option is not true, if $\overrightarrow{\mathrm{A}} = 3\hat{\mathrm{i}} + 4\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = 6\hat{\mathrm{i}} + 8\hat{\mathrm{j}}$, where \(\mathrm{A}\) and \(\mathrm{B}\) are the magnitudes of $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$?
1. $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}} = \overrightarrow{0}$

2. $\frac{\mathrm{A}}{\mathrm{B}} = \frac{1}{2}$

3. $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}} = 48$

4. \(\mathrm{A}=5\)

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

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

Given are two vectors,  $\overrightarrow{\mathrm{A}}= (2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{B}} = (4\hat{\mathrm{i}} - 10\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}})$. What should be the value of c so that vector $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ would become parallel to each other?

1.  1

2.  2

3.  3

4.  4

The angle between vectors $\left(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}\right)$ and $\left(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{A}}\right)$ is

1. Zero

2. $\mathrm{\pi }$

3. $\mathrm{\pi }/4$

4. $\mathrm{\pi }/2$