$\mathrm{If} \overline{\mathrm{a}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}} \mathrm{respectively}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{length} \mathrm{of} \overline{\mathrm{AB}} ?$

• 106

•  - 106

• 53

•  - 53

If the position vectors of points A and B are $3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} +\hspace{0.17em}4\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ respectively, then what is the  AB ? 

• $\sqrt{14}$

• $\sqrt{29}$

• $\sqrt{43}$

• $\sqrt{53}$

If in a right-angled triangle ABC, hypotenuse AC = p, then what is $\overrightarrow{\mathrm{AB}} . \overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{BC}} . \overrightarrow{\mathrm{BA}} + \overrightarrow{\mathrm{CA}} . \overrightarrow{\mathrm{CB}} = ?$

• p²

• 2p²

• $\frac{{\mathrm{p}}^2}{2}$

• p

The sine of the angle between vectors $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 6\hat{\mathrm{j}} - 3\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{is} :$

• $\frac{1}{\sqrt{26}}$

• $\frac{5}{\sqrt{26}}$

• $\frac{5}{26}$

• $\frac{1}{26}$

5.

What is the value of A for which the vectors $3\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} - 2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + 10\hat{\mathrm{k}} \mathrm{are} \mathrm{perpendicular} ?$

• 1

• 2

• 3

• 4

Let the coordinates of the points A, B, C be (1, 8, 4), (0, - 11, 4) and (2. - 3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC ?

• (3, 4, - 2)

• (4, -2, 5)

• (4, 5, - 2)

• (2, 4, 5)

The coordinates of the vertices P, Q and R of a triangle PQR are (1, - 1, 1), (3, - 2, 2) and (0, 2, 6) respectively.If $\angle \mathrm{ROP} = \mathrm{\theta },$ then what is $\angle$PRQ equal to ?

• $30° + \mathrm{\theta }$

• $45° - \mathrm{\theta }$

• $60° - \mathrm{\theta }$

• $90° - \mathrm{\theta }$

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{vectors} \mathrm{such} \mathrm{that} \mathrm{I}\overrightarrow{\mathrm{a}}\mathrm{I} = 2, \mathrm{IbI} = 7 \mathrm{and} \overrightarrow{\mathrm{a}} \mathrm{x} \overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} +6\hat{\mathrm{k}}, \\ \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{acute} \mathrm{angle} \mathrm{between} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} ? \end{gathered}$$

• 30⁰

• 45⁰

• 60⁰

• 90⁰

Let $\overrightarrow{\mathrm{p}} \mathrm{and} \overrightarrow{\mathrm{q}}$ be the position vectors of the points P and Q respectively with respect to origin O. The points R and S divide PQ internally and externally respectively in the ratio 2 : 3. If $\overrightarrow{\mathrm{OR}} \mathrm{and} \overrightarrow{\mathrm{OS}}$ are perpendicular, then which one of the following is correct ?

• 9p² = 4q²

• $4{\mathrm{p}}^2 = 9{\mathrm{q}}^2$

• 9p = 4q

• 4p = 9q

What is the moment about the point $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ of aforce represented by $3\hat{\mathrm{i}} + \hat{\mathrm{k}}$ acting through the point $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ ?

• $- 3\hat{\mathrm{i}} +11\hat{\mathrm{j}} + 9\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} +2\hat{\mathrm{j}} + 9\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 9\hat{\mathrm{k}}$

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