For given vectors, $\vec a = 2\hat i - \hat j + 2\hat k$ and $\vec b = - \hat i + \hat j - \hat k$,

find the unit vector in the direction of the vector $\vec a + \vec b$.

We have, $\vec a = 2\hat i - \hat j + 2\hat k$ and $\vec b = - \hat i + \hat j - \hat k$

$\therefore$ $\vec a + \vec b = (2\hat i - \hat j + 2\hat k) + ( - \hat i + \hat j - \hat k) = \hat i + 0\hat j + \hat k = \hat i + \hat k$

$\therefore$ $|\vec a + \vec b| = \sqrt {{{(1)}^2} + {{(0)}^2} + {{(1)}^2}} = \sqrt {1 + 0 + 1} = \sqrt 2$
$\therefore$

Unit vector in the direction of $\vec a + \vec b$
$= \cfrac{1}{{|\vec a + \vec b|}}(\vec a + \vec b) = \cfrac{1}{{\sqrt 2 }}(\hat i + \hat k) = \cfrac{1}{{\sqrt 2 }}\hat i + \cfrac{1}{{\sqrt 2 }}\hat k$