Vector in the direction of $${L_1} = \overrightarrow {{n_1}} = 3\widehat i + \widehat j + 2\widehat k$$

Vector in the direction of $${L_2} = \overrightarrow {{n_2}} = \widehat i + 2\widehat j + 3\widehat k$$

$$\therefore$$ Vector perpendicular to both L$$_1$$ and L$$_2$$

$$\overrightarrow {{n_1}} \times \overrightarrow {{n_2}} $$

$$ = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & 1 & 2 \cr 1 & 2 & 3 \cr } } \right|$$

$$ = - \widehat i - 7\widehat j + 5\widehat k$$

$$\therefore$$ Required unit vector

$$ = \widehat n = {{ - \widehat i - 7\widehat j + 5\widehat k} \over {\sqrt {1 + 49 + 25} }} = {{ - \widehat i - 7\widehat j + 5\widehat k} \over {5\sqrt 3 }}$$