If $\overrightarrow a = \hat i + \widehat j + \hat k,\overrightarrow b = 2\hat i - \hat j + 3\hat k$ and $\overrightarrow c = \hat i - 2\widehat j + \hat k$,
find a unit vector parallel to the vector$2\overrightarrow a - \overrightarrow b + 3\overrightarrow c$.

$2\overrightarrow a - \vec b + 3\overrightarrow c = 2(\hat i + \widehat j + \hat k) - (2\hat i - \widehat j + 3\hat k) + 3(\hat i - 2\hat j + \hat k)$

$= (2\hat i + 2\hat j + 2\hat k) + ( - 2\hat i + \hat j - 3\hat k) + (3\hat i - 6\hat j + 3\hat k)$

$= (2\hat i - 2\hat i + 3\hat i) + (2\hat j + \hat j - 6j) + (2\hat k - 3\hat k + 3\hat k) = (3\hat i - 3j + 2\hat k)$

$\therefore$

Required unit vector $= \cfrac{{3\hat i - 3\hat j + 2\hat k}}{{\sqrt {9 + 9 + 4} }} = \cfrac{1}{{\sqrt {22} }}(3\hat i - 3\widehat j + 2\hat k)$