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

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• 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 $\overline{{\mathrm{\beta }}_1} \times \overline{{\mathrm{\beta }}_2}$

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

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

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

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

The distance of the point having position vector $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ from the straight line passing  through the point (2, 3, - 4) and parallel to the vector $6\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ is :

• $2\sqrt{13}$

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• $4\sqrt{3}$

517.

A force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} - \hat{\mathrm{j}}$ acts on a point R (0, 1, 1), then the moment of a force about the point P(0, 1, 0) is

• $3\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}}$

• $- \hat{\mathrm{i}} -\hspace{0.17em}3\hat{\mathrm{j}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

Let $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are non-zero and non-collinear vectors. If there exists scalars $\mathrm{\alpha }, \mathrm{\beta }$ such that $\mathrm{\alpha }\overrightarrow{\mathrm{a}} + \mathrm{\beta }\overrightarrow{\mathrm{b}} = \overrightarrow{0}$, then

• $\mathrm{\alpha } = \mathrm{\beta } \ne 0$

• $\mathrm{\alpha } + \mathrm{\beta } = 0$

• $\mathrm{\alpha } = \mathrm{\beta } = 0$

• $\mathrm{\alpha } \ne \mathrm{\beta }$

If G is centroid of $∆$ABC, then

• $\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

• $3\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $3\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be vectors with magnitudes 3, 4 and 5 respectively and $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}$ is

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