Find the unit vector in the direction of the vector $\overrightarrow{PQ}$ where $P (1,2,3)$ and $Q(4,5,6) $.

The correct answer is $\dfrac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$

Given $\overrightarrow{OP}$ = $\hat{i} + 2\hat{j} + 3\hat{k}$ and $\overrightarrow{OQ}$ = $4\hat{i} + 5\hat{j} + 6\hat{k}$

∴ $\overrightarrow{PQ}$ = $\overrightarrow{OQ}$ - $\overrightarrow{OP}$ = $3\hat{i} + 3\hat{j} + 3\hat{k}$

The unit vector in the direction of the vector $\overrightarrow{PQ}$

$\overrightarrow{PQ}$ = $\dfrac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}$ =
$\dfrac{3\hat{i} + 3\hat{j} + 3\hat{k}}{\sqrt{9 + 9 + 9}}$ =
$\dfrac{1}{3\sqrt{3}} (3\hat{i} + 3\hat{j} + 3\hat{k})$

$\dfrac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$