If $\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = \overrightarrow{\mathrm{a}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \overrightarrow{\mathrm{a}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 1, then  $\overrightarrow{\mathrm{a}}$ is equal to

• $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

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

• $\hat{\mathrm{i}}$

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

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are unit vectors and $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = 1$, then $\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|$ is equal to

• $\sqrt{2}$

• 1

• $\sqrt{5}$

• $\sqrt{3}$

The projection of $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{on} \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ is

• $\frac{8}{\sqrt{35}}$

• $\frac{8}{\sqrt{39}}$

• $\frac{8}{\sqrt{14}}$

• $\sqrt{14}$

If $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} = \left|\overrightarrow{\mathrm{a}}\right|\left|\overrightarrow{\mathrm{b}}\right|$, then the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is

• 45°

• 180°

• 90°

• 60°

If $\overrightarrow{\mathrm{a}} +\hspace{0.17em}2\overrightarrow{\mathrm{b}} +\hspace{0.17em}3\overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ is equal to

• $2\left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$

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

• $\overrightarrow{0}$

• $6\left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$

If the volume of the parallelopiped with $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ as coterminous edges is 40 cu unit, then the volume of the parallelopiped having $\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$ as coterminous edges in cubic unit is

• 80

• 120

• 160

• 40

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are non-zero coplanar vectors, then $\left[2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} 3\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}} 4\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right]$ is

• 25

• 0

• 27

• 9

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are unit vectors, such that $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $3\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}= \overrightarrow{0}$ is

• - 1

• 1

• - 3

• 3

If $\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ are unit vectors along the positive direction of x, y and z - axes, then a false statement in the following is

• $\sum \hat{\mathrm{i}} \times \left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} \times \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} . \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} . \left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \overrightarrow{0}$

If a, b and c are unit vectors such that a + b + c = 0, then angle between a and b is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\mathrm{\pi }$