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

1.  1

2.  $\sqrt{65}$

3.  8

4.  4 

$\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors given by $\overrightarrow{\mathrm{A} }= 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. The component of $\stackrel{\to \; }{\mathrm{A}}$parallel to $\overrightarrow{\mathrm{B}}$ is:

1.  $\frac{1}{2}\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

2.  $\frac{5}{2}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

3.  $\frac{5}{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

4.  $\frac{\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}}\right)}{2}$

If a vector is inclined at angles, $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{and} \mathrm{\gamma }$ with x, y, and z-axis respectively, then the value of ${\sin }^2 \mathrm{\alpha } + {\sin }^2 \mathrm{\beta }+ {\sin }^2 \mathrm{\gamma }$ 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 $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors inclined to each other at an angle $\theta$, then the component of $\overrightarrow{\mathrm{A}}$ perpendicular to $\overrightarrow{\mathrm{B}}$ and lying in the plane containing $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ will be:

1.  $\frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

2.  $\overrightarrow{\mathrm{A}} - \frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

3.  $\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}$

4.  $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$

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$°$

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,  $\overrightarrow{\mathrm{A}}= (2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}})$ 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

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

1. $\overrightarrow{B}$ must be perpendicular to $\overrightarrow{C}$

2. $\overrightarrow{A}$ must be perpendicular to $\overrightarrow{C}$

3. Component of $\overrightarrow{C}$ along $\overrightarrow{A}$ = Component of $\overrightarrow{D}$ along $\overrightarrow{A}$

4. Component of $\overrightarrow{C}$ along $\overrightarrow{A}$ = - (Component of $\overrightarrow{D}$ along $\overrightarrow{A}$) 

What is the maximum value of 5sinθ – 12cosθ?

1.  12

2.  17

3.  7

4.  13