If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are perpendicular  to $\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$ respectively and, if   $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|$ = 6, $\left|\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ = 8 and $\left|\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right|$ = 10, then $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to :

• 5$\sqrt{2}$

• 50

• 10$\sqrt{2}$

• 10

A vector perpendicular to $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and coplanar with $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is :

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

• $\hat{\mathrm{i}} + 7\hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $5\left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{a}} = 4\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, then p is :

• 0

• - 1

• 1

• 2

164.

Let $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{k}} - \hat{\mathrm{i}}$. If $\overrightarrow{\mathrm{d}}$ is a unit vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{d}} = 0 = \left[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{d}}\right]$, then $\overrightarrow{\mathrm{d}}$ is (are) :

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

• $\pm \frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}}{\sqrt{6}}$

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

• $\pm \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are unit vectors such that $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right] = \frac{1}{4}$, then angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

If $\overrightarrow{\mathrm{A}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{B}} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{C}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}$, then $\overrightarrow{\mathrm{A}} + \mathrm{t}\overrightarrow{\mathrm{B}}$ is perpendicular to $\overrightarrow{\mathrm{C}}$, if t is equal to :

• - 5

• 4

• 5

• - 4

The magnitude of the projection of the vector $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on the vector perpendicular to the plane containing the vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is

• $\frac{\sqrt{3}}{2}$

• $3\sqrt{6}$

• $\sqrt{6}$

• $\sqrt{\frac{3}{2}}$

Let $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \mathrm{x}\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ , for some real x. Then $\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right| = \mathrm{r}$ is possible if :

• $0 < \mathrm{r} \le \sqrt{\frac{3}{2}}$

• $\sqrt{\frac{3}{2}} < \mathrm{r} \le 3\sqrt{\frac{3}{2}}$

• $3\sqrt{\frac{3}{2}} < \mathrm{r} \le 5\sqrt{\frac{3}{2}}$

• $\mathrm{r} \ge 5\sqrt{\frac{3}{2}}$

If unit vector $\overrightarrow{\mathrm{a}}$ makes angles $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ with $\hat{\mathrm{i}}, \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, with $\hat{\mathrm{j}}$ and $\mathrm{\theta } \in \left(0, \mathrm{\pi }\right)$ with $\hat{\mathrm{k}}$, then a value of $\mathrm{\theta }$ is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{4}$

Let $\overline{\mathrm{\alpha }} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \mathrm{\beta } = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\overline{\mathrm{\beta }} = \overline{{\mathrm{\beta }}_1} - \overline{{\mathrm{\beta }}_2}$, where $\overline{{\mathrm{\beta }}_1}$ is paralle to $\overline{\mathrm{\alpha }}$ and $\overline{{\mathrm{\beta }}_2}$ is perpendicular to $\overline{\mathrm{\alpha }}$, then $\overline{{\mathrm{\beta }}_1} \times \overline{{\mathrm{\beta }}_2}$

• $\frac{1}{2}\left(- 3\overline{\mathrm{i}} + 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}\right)$

• $\frac{1}{2}\left(3\overline{\mathrm{i}} - 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}\right)$

• $3\overline{\mathrm{i}} - 9\overline{\mathrm{j}} - 5\overline{\mathrm{k}}$

• $- 3\overline{\mathrm{i}} + 9\overline{\mathrm{j}} + 5\overline{\mathrm{k}}$