If vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}}$ are coplanar, then $\mathrm{\lambda }$ is equal to

• - 2

• 3

• 2

• - 3

Given, $\overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$, $\left|\overrightarrow{\mathrm{a}}\right| = 1$ and if $\left(\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}}\right)\left(2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) = - 10$, $\left|\overrightarrow{\mathrm{b}}\right|$ is equal to

• 1

• 3

• 2

• 4

$\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right] = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$, then

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right] = 1$

• $\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$ are coplanar

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right] = - 1$

• $\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$ are mutually perpendicular

Area of rhombus is ..., where diagonals are $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = - \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

• $\sqrt{21.5}$

• $\sqrt{31.5}$

• $\sqrt{28.5}$

• $\sqrt{38.5}$

$\frac{\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)}{\overrightarrow{\mathrm{b}} . \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right)} + \frac{\overrightarrow{\mathrm{b}} . \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}{\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \stackrel{ \to }{\mathrm{c}}\right)}$ is equal to

• 1

• 2

• 0

• $\infty$

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 $\left(3\overrightarrow{\mathrm{a}} - 4\overrightarrow{\mathrm{b}}\right) . \left(2\overrightarrow{\mathrm{a}} +\hspace{0.17em}5\overrightarrow{\mathrm{b}}\right)$ is

• - 21

• $- \frac{21}{2}$

• 21

• $\frac{21}{2}$

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 [$\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$] 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 $\overrightarrow{\mathrm{b}} +\hspace{0.17em}\overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{c}} +\hspace{0.17em}\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\overrightarrow{\mathrm{b}}$ 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

529.

If the vectors $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{\lambda }\overrightarrow{\mathrm{b}} +\hspace{0.17em}3\overrightarrow{\mathrm{c}}$, $- 2\overrightarrow{\mathrm{a}} +\hspace{0.17em}3\overrightarrow{\mathrm{b}} -\hspace{0.17em}4\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} -\hspace{0.17em}3\overrightarrow{\mathrm{b}} +\hspace{0.17em}5\overrightarrow{\mathrm{c}}$ are coplanar, then the value of $\mathrm{\lambda }$ is

• 2

• - 1

• 1

• - 2

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, $\left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5, \left|\overrightarrow{\mathrm{c}}\right| = 7$, then anle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$