If the vectors $\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}}$ represents the diagonals of a parallelogram, then its area will be

• $\sqrt{21}$

• $\frac{\sqrt{21}}{2}$

• $2\sqrt{21}$

• $\frac{\sqrt{21}}{4}$

The position vector of the points A, B, C are $\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$, $\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and $\left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ respectively. These points

• form an isosceles triangle

• form a right angled triangle

• are collinear

• form a scalene triangle

If $\left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 3$ and $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ are mutually perpendicular, then the area of the triangle whose vertices are $\overrightarrow{0}, \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}$ is

• 5

• 1

• 6

• 8

Given $\overrightarrow{\mathrm{p}} = 3\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}}$, $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{p}} = \mathrm{x}\overrightarrow{\mathrm{a}} + \mathrm{y}\overrightarrow{\mathrm{b}} + \mathrm{z}\overrightarrow{\mathrm{c}}$ then x, y, z are respectively

• $\frac{3}{2}, \frac{1}{2}, \frac{5}{2}$

• $\frac{1}{2}, \frac{3}{2}, \frac{5}{2}$

• $\frac{5}{2}, \frac{3}{2}, \frac{1}{2}$

• $\frac{1}{2}, \frac{5}{2}, \frac{3}{2}$

If $2\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then ratio in which $\overrightarrow{\mathrm{c}}$ divides $\overrightarrow{\mathrm{AB}}$ is

• 3 : 2 internally

• 3 : 2 externally

• 2 : 3 internally

• 2 : 3 externally

196.

Volume of the parallelopiped having vertices at 0 = (0, 0, 0), A = (2, - 2, 1), B = (5, - 4, 4) and C = (1, - 2, 4) is

• 5 cu unit

• 10 cu unit

• 15 cu unit

• 20 cu unit

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three non-coplanar vectors and $\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{r}}$ are defined by the relations

$\overrightarrow{\mathrm{p}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}, \overrightarrow{\mathrm{q}} = \frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]} \mathrm{and} \overrightarrow{\mathrm{r}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}$

then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{r}}$ is equl to

• 0

• 1

• 2

• 3

The volume of a parallelopiped whose coterminous edges are  $2\overrightarrow{\mathrm{a}}, \overrightarrow{2\mathrm{b}}, 2\overrightarrow{\mathrm{c}}$ is

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

• $4\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $8\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

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

The position vectors of vertices of a $∆$ABC are $4\hat{\mathrm{i}} - 2\hat{\mathrm{j}}$, $\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $- \hat{\mathrm{i}} + 5\hat{\mathrm{j}}+ \hat{\mathrm{k}}$ respectively, then $\angle$ABC is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

If u = a - b and v = a + b and $\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$ = 2, then $\left|\mathrm{u} \times \mathrm{v}\right|$ is equal to

• $2\sqrt{16 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $\sqrt{16 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $\mathrm{2}\sqrt{4 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $2\sqrt{4 + {\left(\mathrm{a} . \mathrm{b}\right)}^2}$