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

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

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  $\overrightarrow{\mathrm{r}}=4\hat{\mathrm{j}}-3\hat{\mathrm{k}},<mspace\; width="1em"></mspace>\mathrm{then\; value\; of}<mspace\; width="1em"></mspace>\left|\overrightarrow{\mathrm{v}}\right|$ will be:

1. $\sqrt{29}<mspace\; width="1em"></mspace>\mathrm{units}$

2. $31<mspace\; width="1em"></mspace>\mathrm{units}$

3. $\sqrt{37}<mspace\; width="1em"></mspace>\mathrm{units}$

4. $\sqrt{41}<mspace\; width="1em"></mspace>\mathrm{units}$

If $\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\hat{\mathrm{j}}$, then $\left|\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{b}}\right|=?$ 

If \(\left|\overrightarrow A\right|\ne \left|\overrightarrow B\right|\) and \(\left|\overrightarrow A \times \overrightarrow B\right|= \left|\overrightarrow A\cdot \overrightarrow B\right|\), then: 

The scalar and vector product of two vectors, $\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right)$ 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 $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>4\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>6\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>8\hat{\mathrm{j}}$, where \(\mathrm{A}\) and \(\mathrm{B}\) are the magnitudes of $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$?
1. $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\overrightarrow{0}$

2. $\frac{\mathrm{A}}{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{1}{2}$

3. $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>48$

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

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

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

Given are two vectors, \(\overrightarrow{A} =   \left(\right. 2 \hat{i}   -   5 \hat{j}   +   2 \hat{k} \left.\right)\) and \(\overrightarrow{B} =   \left(4 \hat{i}   -   10 \hat{j}   +   c \hat{k} \right).\) What should be the value of \(c\) so that vector \(\overrightarrow A \) and \(\overrightarrow B\) would becomes 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. $\mathrm{\pi }/2$