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$\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are perpendicular  to $\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$ respectively and, if   $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|$ = 6, $\left|\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ = 8 and $\left|\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right|$ = 10, then $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to :

• 5$\sqrt{2}$

• 50

• 10$\sqrt{2}$

• 10

507.

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

• $5\left(\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\hat{\mathrm{i}} + 7\hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $5\left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{a}} = 4\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, then p is :

• 0

• - 1

• 1

• 2

Let $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{k}} - \hat{\mathrm{i}}$. If $\overrightarrow{\mathrm{d}}$ is a unit vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{d}} = 0 = \left[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{d}}\right]$, then $\overrightarrow{\mathrm{d}}$ is (are) :

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}}{\sqrt{6}}$

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\pm \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are unit vectors such that $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right] = \frac{1}{4}$, then angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$