If $\overrightarrow {\rm{a}} = \widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$, then find the unit vector in the direction of

(i) $6\overrightarrow {\rm{b}}$

(ii) $2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}}$

Here, $\overrightarrow {\rm{a}} = \widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} + \widehat {\rm{j}} - 2\widehat {\rm{k}}$

(i) Since, $6\overrightarrow {\rm{b}} = 12\widehat {\rm{i}} + 6\widehat {\rm{j}} - 12\widehat {\rm{k}}$

$\therefore$

Unit vector in the direction of $6\overrightarrow {\rm{b}} = \frac{{6\overrightarrow {\rm{b}} }}{{|6\overrightarrow {\rm{b}} |}}$

$= \frac{{12\widehat {\rm{i}} + 6\widehat {\rm{j}} - 12\widehat {\rm{k}}}}{{\sqrt {{{12}^2} + {6^2} + {{12}^2}} }} = \frac{{6(2\widehat {\rm{i}} + \widehat {\rm{j}} - 2\widehat {\rm{k}})}}{{\sqrt {324} }}$

$= \frac{{6(2\widehat {\rm{i}} + \widehat {\rm{j}} - 2\widehat {\rm{k}})}}{{18}} = \frac{{2\widehat {\rm{i}} + \widehat {\rm{j}} - 2\widehat {\rm{k}}}}{3}$

(ii) Since, $2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} = 2(\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}) - (2\widehat {\rm{i}} + \widehat {\rm{j}} - 2\widehat {\rm{k}})$

$= 2\widehat {\rm{i}} + 2\widehat {\rm{j}} + 4\widehat {\rm{k}} - 2\widehat {\rm{i}} - \widehat {\rm{j}} + 2\widehat {\rm{k}} = \widehat {\rm{j}} + 6\widehat {\rm{k}}$

$\therefore$

Unit vector in the direction of $2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} = \frac{{2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} }}{{|2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} |}}$

$= \frac{{\widehat {\rm{j}} + 6\widehat {\rm{k}}}}{{\sqrt {1 + 36} }} = \frac{1}{{\sqrt {37} }}(\widehat {\rm{j}} + 6\widehat {\rm{k}})$