491.

If the volume of a parallelopiped, whose conterminous edges are given by the vectors $$\begin{gathered} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{n}\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} +\hspace{0.17em}4\hat{\mathrm{j}} - \mathrm{n}\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \mathrm{n}\hat{\mathrm{j}} + 3\hat{\mathrm{k}} \left(\mathrm{n} \ge 0\right), \\ \mathrm{is} 158 \mathrm{cu}-\mathrm{units}, \mathrm{then} :\hspace{0.17em} \end{gathered}$$

• n = 7

• n = 9

• $\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} = 10$

• $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}} = 17$

$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{vectors}\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}} \mathrm{be} \mathrm{such} \mathrm{that} \left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 4 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 4. \mathrm{If} \mathrm{the} \mathrm{projection} \mathrm{of} \\ \overrightarrow{\mathrm{b}} \mathrm{on} \overrightarrow{\mathrm{a}} \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{the} \mathrm{projection} \mathrm{of} \overrightarrow{\mathrm{c}}\mathrm{on} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{is} \mathrm{perpendicular} \mathrm{to} \overrightarrow{\mathrm{c}}, \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|\mathrm{is} ....... \end{gathered}$$

$\mathrm{If} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{unit} \mathrm{vectors}, \mathrm{then} \mathrm{the} \mathrm{greatest} \mathrm{value} \mathrm{of} \sqrt{3}\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| + \left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right| = ?$