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

• 3

• $\sqrt{3}$

• $\left(\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2}\right)/3$

• 1

If$\overrightarrow{\mathrm{p}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}, \overrightarrow{\mathrm{q}} = 4\hat{\mathrm{k}} - \hat{\mathrm{j}} \mathrm{and} \overrightarrow{\mathrm{r}} = \hat{\mathrm{i}} + \hat{\mathrm{k}}$ then the unit vector in the direction of $3\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}} - 2\overrightarrow{\mathrm{r}}$ is :

• $\frac{1}{3}\left(\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} -\hspace{0.17em}2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} -\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are the two vectors such that $\left|\overrightarrow{\mathrm{a}}\right| = 3\sqrt{3}, \left|\overrightarrow{\mathrm{b}}\right| = 4$ and $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = \sqrt{7}$, then the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $120°$

• $60°$

• $30°$

• $150°$

If $\overrightarrow{\mathrm{a}}$ is vector perpendicular to both $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, then :

• $\overrightarrow{\mathrm{a}} + \left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

315.

If the area of the parallelogram with $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ as two adjacent sides is 15 sq unit, then the area of the parallelogram having $3\overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}}$ as two adjacent sides in sq unit is :

• 120

• 105

• 75

• 45

If $\mathrm{a} = 2\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} - 5\hat{\mathrm{k}}, \mathrm{c} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - \hat{\mathrm{k}}$, then a vector perpendicular to a and in the plane containing b and c is

• $- 17\hat{\mathrm{i}} + 21\hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $17\hat{\mathrm{i}} + 21\hat{\mathrm{j}} - 123\hat{\mathrm{k}}$

• $- 17\hat{\mathrm{i}} - 21\hat{\mathrm{j}} + 97\hat{\mathrm{k}}$

• $- 17\hat{\mathrm{i}} - 21\hat{\mathrm{j}} -97 \hat{\mathrm{k}}$

OA and BO are two vectors of magnitudes 5 and 6 respectively. If $\angle$BOA = 60°, then OA · OB is equal to

• 0

• 15

• - 15

• $15\sqrt{3}$

A vector perpendicularto the plane containing the points A (1, - 1, 2), B(2, 0, - 1), C(0, 2, 1) is

• $4\hat{\mathrm{i}} + 8\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$

• $8\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

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

If a and b are vectors such that $\left|\mathrm{a} + \mathrm{b}\right| = \left|\mathrm{a} - \mathrm{b}\right|$, then the angle between a and b is

• 120°

• 60°

• 90°

• 30°

A unit vector perpendicular to both the vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{j}} + \hat{\mathrm{k}}$

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

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

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

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