$\left(\mathrm{a} . \hat{\mathrm{i}}\right)\hat{\mathrm{i}} + \left(\mathrm{a} . \hat{\mathrm{j}}\right)\hat{\mathrm{j}} +\hspace{0.17em}\left(\mathrm{a} . \hat{\mathrm{k}}\right)\hat{\mathrm{k}}$ is equal to

• a

• 2a

• 3a

• 0

If $\left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4$, then a value of $\mathrm{\lambda }$ for which a + $\mathrm{\lambda b}$ is perpendicular to a - $\mathrm{\lambda b}$ is

• $\frac{9}{16}$

• $\frac{3}{4}$

• $\frac{3}{2}$

• $\frac{4}{3}$

The projection of $\mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ on $\mathrm{b} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is

• $\frac{1}{\sqrt{14}}$

• $\frac{2}{\sqrt{14}}$

• $\sqrt{14}$

• $\frac{- 2}{\sqrt{14}}$

If the vectors $4\hat{\mathrm{i}} + 11\hat{\mathrm{j}} + \mathrm{m}\hat{\mathrm{k}}$, $7\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ are coplanar, then m is equal to

• 0

• 38

• - 10

• 10

If $\left|\mathrm{a} \times \mathrm{b}\right| = 4 \mathrm{and} \left|\mathrm{a} . \mathrm{b}\right| = 2, \mathrm{then} {\left|\mathrm{a}\right|}^2{\left|\mathrm{b}\right|}^2$ is equal to

• 6

• 2

• 20

• 8

The angle between the vectors a + b and a - b when a = (1, 1, 4) and b = (1, - 1, 4) is

• 45°

• 90°

• 15°

• 30°

If a, b, c are mutually perpendicular unit vectors, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• 3

• $\sqrt{3}$

• 0

• 1

If ABCDEF is a regular hexagon, then AD + EB + FC equals

• 0

• 2 AB

• 4 AB

• 3 AB

309.

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

• $- \frac{3}{\sqrt{14}}$

• $\frac{3}{\sqrt{14}}$

• $- \sqrt{\frac{3}{2}}$

• $\frac{3}{\sqrt{2}}$

An unit vector perpendicular to the plane containing the vectors $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is

• $\pm \frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{k}}}{\sqrt{2}}$

• $\pm \frac{\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{2}}$

• $\mp \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$