Unit 13.1 Worked Example 1

The position vectors of two points $A$ and $B$ are $\left( 3\mathbf{i}-\mathbf{j}+4\mathbf{k} \right)$ and $\left( 2\mathbf{i}+\mathbf{j}+2\mathbf{k} \right)$ respectively. Points $A$, $B$ and $C$ are collinear. Given that point $C$ is $6$ units away from $A$ , find the possible position vectors of $C$.