Two vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ of equal magnitude 5 originating from a point and directs respectively towards north-east and north-west. Then the magnitude of $\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}$ is :

• $3\sqrt{2}$

• $2\sqrt{3}$

• $2\sqrt{5}$

• $5\sqrt{2}$

ABCD is a quadrilateral, P, Q are the mid points of $\overrightarrow{\mathrm{BC}} \mathrm{and} \overrightarrow{\mathrm{AD}}$, then $\overrightarrow{\mathrm{AB}} + \overrightarrow{\mathrm{DC}}$ is equal to :

• 3$\overrightarrow{\mathrm{QP}}$

• $\overrightarrow{\mathrm{QP}}$

• $4\overrightarrow{\mathrm{QP}}$

• $2\overrightarrow{\mathrm{QP}}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are are the three vectors mutually perpendicular to each other and $\left|\overrightarrow{\mathrm{a}}\right| = 1, \left|\overrightarrow{\mathrm{b}}\right| = 3 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 5$, then $\left|\overrightarrow{\mathrm{a}} - 2\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}} - 3\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} - 4\overrightarrow{\mathrm{a}}\right|$ is equal to

• 0

• - 24

• 3600

• - 215

154.

A unit vector coplanar with $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$, and perpendicular to $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

• $\left(\frac{\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{2}}\right)$

• $\left(\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}\right)$

• $\left(\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}}{\sqrt{6}}\right)$

• $\left(\frac{\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{6}}\right)$

Forces acting on a particle have magnitude 5, 3 and 1 unit and act in the direction of the vectors $6\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$ respectively. They remain constant while the particle is displaced from the point A (2, - 1, - 3) to B (5, - 1, 1). The work done is :

• 11 unit

• 33 unit

• 10 unit

• 30 unit

If $\left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 1 \mathrm{and} \left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = \sqrt{3}$, then the value of $\left(3\overrightarrow{\mathrm{a}} - 4\overrightarrow{\mathrm{b}}\right) . \left(2\overrightarrow{\mathrm{a}} + 5\overrightarrow{\mathrm{b}}\right)$ is :

• - 21

• $- \frac{21}{2}$

• 21

• $\frac{21}{2}$

If $\left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 4, \left|\overrightarrow{\mathrm{c}}\right| = 5 \mathrm{and} \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are such that each is perpendicular to the sum of other two, then $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is :

• 5$\sqrt{2}$

• $\frac{5}{\sqrt{2}}$

• $\mathrm{10}\sqrt{2}$

• $10\sqrt{3}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are unit vectors, then  $\left|2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, 2\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}, 2\mathrm{c} - \overrightarrow{\mathrm{a}}\right|$ is equal to :

• 1

• 0

• $- \sqrt{3}$

If $\overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}}, \overrightarrow{\mathrm{w}}$ be vectors such that $\overrightarrow{\mathrm{u}} + \overrightarrow{\mathrm{v}} + \overrightarrow{\mathrm{w}} = \overrightarrow{0}$  and $\left|\overrightarrow{\mathrm{u}}\right| = 3, \left|\overrightarrow{\mathrm{v}}\right| = 4, \left|\overrightarrow{\mathrm{w}}\right| = 5$, then $\overrightarrow{\mathrm{u}} . \overrightarrow{\mathrm{v}} + \overrightarrow{\mathrm{v}} . \overrightarrow{\mathrm{w}} + \overrightarrow{\mathrm{w}} . \overrightarrow{\mathrm{u}}$ is equal to :

• 47

• - 47

• 0

• - 25

If $\overrightarrow{\mathrm{a}}$ is perpendicular  to $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$,   $\left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 3, \left|\overrightarrow{\mathrm{c}}\right| = 4$ and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{2\mathrm{\pi }}{3}$, then $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$ is equal to :

• $4\sqrt{3}$

• 6$\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$