If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are any three coplanar unit vectors, then

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = 1$

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ = 3

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = 0$

• $\left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) . \overrightarrow{\mathrm{b}} = 1$

If a force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ is acting at the point P(1, - 1, 2), then the moment of $\overrightarrow{\mathrm{F}}$ about the point Q(2, - 1, 3) is :

• $\sqrt{57}$

• $\sqrt{39}$

• 12

• 17

A vector of magnitude 5 and perpendicular to  ($\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$) and $\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ :

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{5\sqrt{3}}{3}\left(- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

294.

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be three vectors. Then, scalar triple product $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$ is equal to :

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

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

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

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

$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are three vectors with magnitude $\left|\overrightarrow{\mathrm{a}}\right| = 4, \left|\overrightarrow{\mathrm{b}}\right| = 4, \left|\overrightarrow{\mathrm{c}}\right| = 2$ and such that $\overrightarrow{\mathrm{a}}$ is perpendicular to $\left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right), \overrightarrow{\mathrm{b}}$, is perpendicular to $\left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right) \mathrm{and} \overrightarrow{\mathrm{c}}$ is perpendicular to $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$. It follows that $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to :

• 9

• 6

• 5

• 4

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three non-coplanar vectors, then

$\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) . \left[\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{c}}\right)\right]$ is :

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

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

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

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{j}}$ are coplanar vectors, then te value of $\mathrm{\alpha }$ is :

• $- \frac{4}{3}$

• $\frac{3}{4}$

• $\frac{4}{3}$

• 2

If the position vectors of the vertices A, B, C of a triangle ABC are $7\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, $- \hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $- 4\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ respectively, the triangle is :

• equilateral

• isosceless

• scalene

• right angled and isosceless also

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three unit vectors such that  $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, where $\overrightarrow{0}$ is null vector, then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} = \overrightarrow{0}$ is :

• - 3

• - 2

• $- \frac{3}{2}$

• 0

The area of the parallelogram whose adjacent sides are $\hat{\mathrm{i}} - \hat{\mathrm{k}}$ and $2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}}$ is :

• 2

• 4

• $\sqrt{17}$

• $2\sqrt{13}$