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 $\left|\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{b}}\right|=?$ 

1.	1	2.	$\sqrt{65}$
3.	8	4.	4

\(\overrightarrow A\) and \(\overrightarrow {B}\) are two vectors given by \(\overrightarrow {A}= 2\hat i + 3\hat j\) and \(\overrightarrow {B}= \hat i + \hat j\). The component of \(\overrightarrow A\) parallel to \(\overrightarrow B\) is:
1. \(\frac{(2\hat i -\hat j)}{2}\)
2. \(\frac{5}{2}(\hat i - \hat j)\)
3. \(\frac{5}{2}(\hat i + \hat j)\)
4. \(\frac{(3\hat i -2\hat j)}{2}\)

If a vector is inclined at angles \(\alpha ,\beta ,~\text{and}~\gamma\), with \(x\), \(y\), and \(z\)-axis respectively, then the value of \(\sin^{2}\alpha+\sin^{2}\beta+ \sin^{2}\gamma\)${}^{}$
is equal to:
1. \(0\)
2. \(1\)
3. \(2\)
4. \(\frac{1}{2}\)

A force of \(20\) N acts on a particle along a direction, making an angle of \(60^\circ\) with the vertical. The component of the force along the vertical direction will be:

If \(\overrightarrow {A}\) $\mathrm{and}$ \(\overrightarrow{B}\) are two vectors inclined to each other at an angle \(\theta,\) then the component of \(\overrightarrow {A}\)$\stackrel{}{}$ perpendicular to \(\overrightarrow {B}\) and lying in the plane containing \(\overrightarrow {A}\) and$$ \(\overrightarrow {B}\) will be:
1. \(\frac{\overrightarrow {A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}\)
2. \(\overrightarrow{A}   -   \frac{\overrightarrow{A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}\)
3. \(\overrightarrow{A} -\overrightarrow{B}\)
4. \(\overrightarrow{A} + \overrightarrow{B}\)$\stackrel{}{}$

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: 

Two forces of the same magnitude are acting on a body in the East and North directions, respectively. If the body remains in equilibrium, then the third force should be applied in the direction of:

1. North-East

2. North-West

3. South-West

4. South-East

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. \(4\)

If \(\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{C} + \overrightarrow{D}\), then which of the following statement is correct?

What is the maximum value of \(5\sin\theta-12\cos\theta\)?

1. \(12\)

2. \(17\)

3. \(7\)

4. \(13\)