The distance of the point having position vector $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ from the straight line passing  through the point (2, 3, - 4) and parallel to the vector $6\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ is :

• $2\sqrt{13}$

• 7

• 6

• $4\sqrt{3}$

172.

A force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} - \hat{\mathrm{j}}$ acts on a point R (0, 1, 1), then the moment of a force about the point P(0, 1, 0) is

• $3\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}}$

• $- \hat{\mathrm{i}} -\hspace{0.17em}3\hat{\mathrm{j}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

Let $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are non-zero and non-collinear vectors. If there exists scalars $\mathrm{\alpha }, \mathrm{\beta }$ such that $\mathrm{\alpha }\overrightarrow{\mathrm{a}} + \mathrm{\beta }\overrightarrow{\mathrm{b}} = \overrightarrow{0}$, then

• $\mathrm{\alpha } = \mathrm{\beta } \ne 0$

• $\mathrm{\alpha } + \mathrm{\beta } = 0$

• $\mathrm{\alpha } = \mathrm{\beta } = 0$

• $\mathrm{\alpha } \ne \mathrm{\beta }$

If G is centroid of $∆$ABC, then

• $\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

• $3\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $3\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be vectors with magnitudes 3, 4 and 5 respectively and $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}$ is

• 47

• 25

• 50

• - 25

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

• $\infty$