Find the unit vectors parallel to $3 \overrightarrow{a} - 2\overrightarrow{b} + 4 \overrightarrow{c}$ where $\overrightarrow{a} = 3 \hat{i} - \hat{j} - 4 \hat{k}$, $\overrightarrow{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}$ and $\overrightarrow{c} = \hat{i} + 2 \hat{j} - \hat{k}$
$\begin{aligned}
\text{Let } \overrightarrow{P} & = 3 \overrightarrow{a} - 2 \overrightarrow{b} + 4 \overrightarrow{c} \\\\
& = 3 \left(3 \hat{i} - \hat{j} - 4 \hat{k}\right) - 2 \left(-2 \hat{i} + 4 \hat{j} - 3 \hat{k}\right) + 4 \left(\hat{i} + 2 \hat{j} - \hat{k}\right) \\\\
& = 9 \hat{i} - 3 \hat{j} - 12 \hat{k} + 4 \hat{i} - 8 \hat{j} + 6 \hat{k} + 4 \hat{i} + 8 \hat{j} - 4 \hat{k} \\\\
& = 17 \hat{i} - 3 \hat{j} - 10 \hat{k}
\end{aligned}$
Let $\hat{p}$ be the unit vector parallel to $\overrightarrow{P}$.
$\begin{aligned}
\text{Then, } \hat{p} & = \pm \left(\dfrac{\overrightarrow{P}}{\left|\overrightarrow{P}\right|}\right) \\\\
& = \pm \left(\dfrac{17 \hat{i} - 3 \hat{j} - 10 \hat{k}}{\sqrt{\left(17\right)^2 + \left(-3\right)^2 + \left(-10\right)^2}}\right) \\\\
& = \pm \left(\dfrac{17 \hat{i} - 3 \hat{j} - 10 \hat{k}}{\sqrt{398}}\right)
\end{aligned}$