If the position vectors of $P$ and $Q$ are $\hat{i} + 3 \hat{j} - 7 \hat{k}$ and $5 \hat{i} - 2 \hat{j} + 4 \hat{k}$, find $\overrightarrow{PQ}$ and determine its direction cosines.
Position vector of point $P = \hat{i} + 3 \hat{j} - 7 \hat{k}$
Position vector of point $Q = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$
$\begin{aligned}
\overrightarrow{PQ} & = \text{position vector of Q} - \text{position vector of P} \\\\
& = \left(5 \hat{i} - 2 \hat{j} + 4 \hat{k}\right) - \left(\hat{i} + 3 \hat{j} - 7 \hat{k}\right) \\\\
& = 4 \hat{i} - 5 \hat{j} + 11 \hat{k}
\end{aligned}$
$\left|\overrightarrow{PQ}\right| = \sqrt{\left(4\right)^2 + \left(-5\right)^2 + \left(11\right)^2} = \sqrt{162} = 9 \sqrt{2}$
$\therefore$ $\;$ The direction cosines are $\;$ $\dfrac{4}{9 \sqrt{2}}$, $\dfrac{-5}{9 \sqrt{2}}$, $\dfrac{11}{9 \sqrt{2}}$