For any vector $\overrightarrow{\mathrm{a}}, \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{j}} \times \left(\stackrel{\to }{\mathrm{a}}\times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ is equal to:

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}}$

• $2\overrightarrow{\mathrm{a}}$

• $3\overrightarrow{\mathrm{a}}$

If the vectors $3\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 8\hat{\mathrm{k}}$ are perpendicular, then $\mathrm{\lambda }$ is :

• - 14

• 7

• 14

• $\frac{1}{7}$

The projection of the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ along the vector $\hat{\mathrm{j}}$ is:

• 1

• 0

• 2

• - 1

The projection of $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ is :

• 1/7

• - 1/7

• 7

• - 7

145.

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = 0$ and $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$ = 0, then :

• $\overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} \parallel \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{and} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{or} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$

${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2$ is equal to :

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

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

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2 - 2\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2 - 2 . \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

Let ABCD be the parallelogram whose sides AB and AD are represented by the vectors $2\overrightarrow{\mathrm{i}} + 4\overrightarrow{\mathrm{j}} - 5\overrightarrow{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{i}} + 2\overrightarrow{\mathrm{j}} - 3\overrightarrow{\mathrm{k}}$ respectively. Then, if a is a unit vector parallel to $\overrightarrow{\mathrm{AC}}$, then $\overrightarrow{\mathrm{a}}$ equals:

• $\frac{1}{3}\left(3\overrightarrow{\mathrm{i}} - 6\overrightarrow{\mathrm{j}} - 2\overrightarrow{\mathrm{k}}\right)$

• $\frac{1}{3}\left(3\overrightarrow{\mathrm{i}} + 6\overrightarrow{\mathrm{j}} + 2\overrightarrow{\mathrm{k}}\right)$

• $\frac{1}{7}\left(3\overrightarrow{\mathrm{i}} - 6\overrightarrow{\mathrm{j}} - 2\overrightarrow{\mathrm{k}}\right)$

• $\frac{1}{7}\left(3\overrightarrow{\mathrm{i}} + 6\overrightarrow{\mathrm{j}} - 2\overrightarrow{\mathrm{k}}\right)$

Let $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ be two unit vectors such that angle between them is 60°. Then $\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|$ is equal to :

• $\sqrt{5}$

• $\sqrt{3}$

• 0

• 1

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right)$ $+ \hat{\mathrm{j}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ then length of $\overrightarrow{\mathrm{b}}$ is equal to :

• $\sqrt{12}$

• $2\sqrt{12}$

• $3\sqrt{14}$

• $2\sqrt{14}$

The velocity of a boat X relative to a boat Y is $5\hat{\mathrm{i}} - 2\hat{\mathrm{j}}$ and that of Y relative to another boat Z is $9\hat{\mathrm{i}} + 4\hat{\mathrm{j}}$ where $\hat{\mathrm{i}} \mathrm{and} \hat{\mathrm{j}}$ are the velocity of k not per hour, east and north respectively. Then the velocity is :

• $\frac{\sqrt{2}}{10} \mathrm{knot}/\mathrm{h}$

• $\frac{10}{\sqrt{2}}$ knot/h

• $10\sqrt{2}$ knot/h

• $2\sqrt{10}$ knot/h