If the vectors $\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{a}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = \hat{\mathrm{j}}$ are such that $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{b}}$ form a right handed system, then $\overrightarrow{\mathrm{c}}$ is

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

• $\overrightarrow{0}$

• $\mathrm{y}\hat{\mathrm{j}}$

• $- \mathrm{z}\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{k}}$

The vector $\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude, then it becomes $4\hat{\mathrm{i}} + \left(4\mathrm{x} - 2\right)\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. Then, the values of x are

• $- \frac{2}{3}, 2$

• $\frac{1}{3}, 2$

• $\frac{2}{3}, 0$

• 2, 7

353.

If a = (1, - 1) and b = (- 2, m) are two collinear vectors, then m is equal to

• 4

• 3

• 2

• 0

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

• 2a

• 3a

• - 2a

• a

If the a, band c form the sides BC, CA and AB respectively, of a $∆$ABC, then

• $\mathrm{a} . \mathrm{b} + \mathrm{b} . \mathrm{c} + \mathrm{c} . \mathrm{a} = 0$

• $\mathrm{a} \times \mathrm{b} = \mathrm{b} \times \mathrm{c} = \mathrm{c} \times \mathrm{a}$

• $\mathrm{a} . \mathrm{b} = \mathrm{b} . \mathrm{c} = \mathrm{c} . \mathrm{a} = 0$

• $\mathrm{a} \times \mathrm{b} + \mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a}$

If a, b and c are three vectors, such that $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right| = 0$, $\left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2, \left|\mathrm{c}\right| = 3$, then a . b + b . c + c . a is equal to

• 0

• - 7

• 7

• 1

If the position vectors of P and Q are $\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 7\hat{\mathrm{k}}\right)$ and $\left(5\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$, then $\left|\mathrm{PQ}\right|$ is

• $\sqrt{158}$

• $\sqrt{160}$

• $\sqrt{161}$

• $\sqrt{162}$

Let u = $\hat{\mathrm{i}} + \hat{\mathrm{j}}$, v = $\hat{\mathrm{i}} - \hat{\mathrm{j}}$ and w = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\hat{\mathrm{n}}$ is a unit vector such that $\mathrm{u} . \hat{\mathrm{n}}$ = 0 and $\mathrm{v} . \hat{\mathrm{n}}$ = 0, then $\left|\mathrm{w} . \hat{\mathrm{n}}\right|$ is equal to

• 0

• 1

• 2

• 3

If A = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$, B = $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and C = $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, then $\left(\mathrm{A} \times \mathrm{B}\right) \times \mathrm{C}$ is

• $5\left(- \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $4\left(- \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $5\left(- \hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$

• $4\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

A particle is acted upon by constant forces $4\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ which displace it from a point $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ to the point $5\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + \hat{\mathrm{k}}$. The work done in standard unit by the forces is given by

• 40

• 30

• 25

• 15