Mathematics : Vector Algebra

${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2$ 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

Two vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ of equal magnitude 5 originating from a point and directs respectively towards north-east and north-west. Then the magnitude of $\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}$ is :

• $3\sqrt{2}$

• $2\sqrt{3}$

• $2\sqrt{5}$

• $5\sqrt{2}$

ABCD is a quadrilateral, P, Q are the mid points of $\overrightarrow{\mathrm{BC}} \mathrm{and} \overrightarrow{\mathrm{AD}}$, then $\overrightarrow{\mathrm{AB}} + \overrightarrow{\mathrm{DC}}$ is equal to :

• 3$\overrightarrow{\mathrm{QP}}$

• $\overrightarrow{\mathrm{QP}}$

• $4\overrightarrow{\mathrm{QP}}$

• $2\overrightarrow{\mathrm{QP}}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are are the three vectors mutually perpendicular to each other and $\left|\overrightarrow{\mathrm{a}}\right| = 1, \left|\overrightarrow{\mathrm{b}}\right| = 3 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 5$, then $\left|\overrightarrow{\mathrm{a}} - 2\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}} - 3\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} - 4\overrightarrow{\mathrm{a}}\right|$ is equal to

• 0

• - 24

• 3600

• - 215

A unit vector coplanar with $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$, and perpendicular to $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

• $\left(\frac{\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{2}}\right)$

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

• $\left(\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}}{\sqrt{6}}\right)$

• $\left(\frac{\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{6}}\right)$

Forces acting on a particle have magnitude 5, 3 and 1 unit and act in the direction of the vectors $6\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$ respectively. They remain constant while the particle is displaced from the point A (2, - 1, - 3) to B (5, - 1, 1). The work done is :

• 11 unit

• 33 unit

• 10 unit

• 30 unit