If the vector $\overline{\mathrm{a}} = 2\overline{\mathrm{i}} + 3\overline{ \mathrm{j}} + 6\overline{\mathrm{k}}$ and $\overline{b}$ are collinear and $\overline{\left|b\right|} = 21$ then $\overline{b}$ =

Given $\overline{\mathrm{a}} = 2\overline{\mathrm{i}} + 3\overline{ \mathrm{j}} + 6\overline{\mathrm{k}}$,  $\overline{\left|b\right|} = 21$

$\overline{a}, \overline{b }$ are collinear vectors

$\overline{\mathrm{b}} = \mathrm{\lambda }\overline{\mathrm{a}} \Rightarrow \overline{\mathrm{b}} = \mathrm{\lambda }\left(2\overline{\mathrm{i}} + 3\overline{\mathrm{j}} + 6\overline{\mathrm{k}}\right)$

$$\begin{gathered} \left|\overline{\mathrm{b}}\right| = \left|\mathrm{\lambda }\right|\sqrt{4+9+36} = 21 \\ \Rightarrow \left|\mathrm{\lambda }\right| . 7 = 21 \Rightarrow \mathrm{\lambda } =\pm 3 \\ \therefore \overline{\mathrm{b}}= \pm 3\left(2\overline{\mathrm{i}} + 3\overline{\mathrm{j}} + 6\overline{\mathrm{k}}\right) \end{gathered}$$