Answer
$\textbf{r}_{separation vector}=(2\hat{x}-2\hat{y}+\hat{z})$
Magnitude of the separation vector:
$|\textbf{r}_{separation vector}|=r_{separation vector}=3$
Unit vector:
$\hat{\textbf{r}}_{separation vector}=\large(\frac{2\hat{x}}{3}-\frac{2\hat{y}}{3}+\frac{\hat{z}}{3})$
Work Step by Step
$\textbf{r}_{separation vector}=(4\hat{x}+6\hat{y}+8\hat{z})-(2\hat{x}+8\hat{y}+7\hat{z}) = (2\hat{x}-2\hat{y}+\hat{z})$
Magnitude of the separation vector:
$|\textbf{r}_{separation vector}|=r_{separation vector}=\sqrt{2^{2}+(-2)^{2}+1^{2}}=\sqrt{4+4+1}=\sqrt{9}=3$
Unit vector:
$\hat{\textbf{r}}_{separation vector}=\large\frac{\textbf{r}_{separation vector}}{|\textbf{r}_{separation vector}|}=\frac{(2\hat{x}-2\hat{y}+\hat{z})}{3}=(\frac{2\hat{x}}{3}-\frac{2\hat{y}}{3}+\frac{\hat{z}}{3})$
