If a and b are the position vectors of A and B, then position vector of point C on AB produced such that AC = 3AB, is

• 3a - b

• 3b - a

• 3a - 2b

• 3b - 2a

The area of a parallelogram whose adjacent sides are $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 4\hat{\mathrm{k}}$, is

• $5\sqrt{3}$

• $10\sqrt{3}$

• $5\sqrt{6}$

• $10\sqrt{6}$

${\left|\mathrm{a} \times \mathrm{b}\right|}^2 + {\left|\mathrm{a} . \mathrm{b}\right|}^2$ is equal to

• $\left(\mathrm{a} \times \mathrm{b}\right) . \left(\mathrm{b} \times \mathrm{b}\right)$

• $\left(\mathrm{a} . \mathrm{a}\right)\left(\mathrm{b} . \mathrm{b}\right)$

• $\left|\mathrm{a} \times \mathrm{b}\right|\left(\mathrm{a} . \mathrm{b}\right)$

• $2\left|\mathrm{a} \times \mathrm{b}\right|\left(\mathrm{a} . \mathrm{b}\right)$

The unit vector prependicular to each of the vectors $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ is

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• None of these

375.

Find the moment of the force $5\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 16\hat{\mathrm{k}}$ acting at the point $2\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, about the point $- 5\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 10\hat{\mathrm{k}}$

• $41\hat{\mathrm{i}} - 8\hat{\mathrm{j}} + 55\hat{\mathrm{k}}$

• $- 408\hat{\mathrm{i}} - 12\hat{\mathrm{j}} + 135\hat{\mathrm{k}}$

• $- 36\hat{\mathrm{i}} + 14\mathrm{j} - 35\hat{\mathrm{k}}$

• None of the above

The vector of magnitude 9 unit perpendicular to the vectors $4\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $- 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ will be

• $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$

• $- 3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} - 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}} \ne \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}$ will be equal to

• $\mathrm{k}\overrightarrow{\mathrm{b}}$

• $\mathrm{k}\overrightarrow{\mathrm{a}}$

• $\mathrm{k}\overrightarrow{\mathrm{c}}$

• $\mathrm{k}\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$

Which one of the following is not defined?

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

• $\overrightarrow{\mathrm{a}} \times 2\overrightarrow{\mathrm{b}}$

• $\left(3\overrightarrow{\mathrm{a}} \times 2\overrightarrow{\mathrm{b}}\right)4\overrightarrow{\mathrm{a}}$

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

The vectors $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ and $\mathrm{a}\hat{\mathrm{i}} + \mathrm{b}\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}}$ are perpendicular, when

• a = 2, b = 3, c = - 4

• a = 4, b = 4, c = 5

• a = 4, b = 4, c = - 5

• None of the above

A man runs due North at the rate of 20 km/h and the wind blows fromthe East at the rate of 15km/h. Then, the velocity of the wind relative to the man will be

• 2km/h

• 3 km/h

• 25 km/h

• None of these