If the vectors $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 4\hat{\mathrm{k}},$ $\overrightarrow{\mathrm{b}} = 4\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - \mathrm{\lambda }\hat{\mathrm{k}}$ are coplanar, then the value of $\mathrm{\lambda }$ is equal to

• 2

• 1

• 3

• - 1

92.

Let A(1, - 1, 2) and B (2, 3, - 1) be two points. If a point P divides AB internally in the ratio 2 : 3, then the position vector of P is

• $\frac{1}{\sqrt{5}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{3}}\left(\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{3}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{5}\left(7\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

If the scalar product of the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ with the unit vector along $\mathrm{m}\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is equal to 2, then one of the values of m is

• 3

• 4

• 5

• 6

The vector equation of the straight line $\frac{1 - \mathrm{x}}{3} = \frac{\mathrm{y} + 1}{- 2} = \frac{3 - \mathrm{z}}{- 1}$ is

• $\overrightarrow{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

If a is perpendicular to b, then the vector $\mathrm{a} \times \left\{\mathrm{a} \times \left\{\mathrm{a} \times \left(\mathrm{a} \times \mathrm{b}\right)\right\}\right\}$ is equal to

• ${\left|\mathrm{a}\right|}^2\mathrm{b}$

• $\left|\mathrm{a}\right|\mathrm{b}$

• ${\left|\mathrm{a}\right|}^3\mathrm{b}$

• ${\left|\mathrm{a}\right|}^4\mathrm{b}$

If the vector 8i + aj of magnitude 10 is in the direction of the vector 4i + 3j, then the value of equal to

• 6

• 3

• - 3

• - 6

If a = 2i - 7j + k and b = i + 3j - 5k and a · mb = 120, then the value of m is equal to

• 5

• - 24

• - 5

• 120

The position vector of the centroid of the $∆$ABC is 2i + 4 j + 2k. If the position vector of the vertex A is 2i + 6j + 4k, then the position vector of midpoint of BC is

• 2i + 3j + k

• 2i + 3j - k

• 2i - 3j - k

• - 2i - 3j - k

If the vectors 3i - 4j - k and 2i + 3j - 6k represent the diagonals of a rhombus, then the length of the side of the rhombus is

• 15

• $15\sqrt{3}$

• $\frac{5\sqrt{3}}{2}$

• $\frac{15\sqrt{3}}{2}$

If a = 2i + 3j + $\mathrm{\alpha }$k and b = 3i - $\mathrm{\alpha }$j + 2k, then the angle between a + b and a-b is equal to

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$