If the vectors $\mathrm{a}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$ are perpendicularto each other, then a is given by

• 9

• 16

• 25

• 36

712.

If position vectors of points A,B,C are $\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ respectively and AB = CX, then position vector of point X is

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

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

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

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

If the position yectors of the points A, B, C, D are $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$, $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $- 5\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, then

• AB = CD

• $\mathrm{AB} \parallel \mathrm{CD}$

• $\mathrm{AB} \perp \mathrm{CD}$

• None of the above

If $\left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4, \left|\mathrm{c}\right| = 5$ are such that each of them is perpendicular to the sum ofthe other two, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• $5\sqrt{2}$

• $\frac{5}{\sqrt{2}}$

• $10\sqrt{2}$

• $5\sqrt{3}$

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

• $\frac{1}{\sqrt{14}}$

• $\frac{2}{\sqrt{14}}$

• $\frac{3}{\sqrt{14}}$

• None of these

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

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