Given the following vector, $\mathbf{v}$, obtain the unit vector that aligns with the direction of $\mathbf{v}$.

$\mathbf{v}=3\mathbf{i}+4\mathbf{k}$

Given the following vector, $\mathbf{v}$, obtain the unit vector that aligns with the direction of $\mathbf{v}$.

$\mathbf{v}=-5\mathbf{i}+2\mathbf{j}$

A vector's magnitude is $8$ and it forms a $120^{\omicron}$ angle with the positive $x$-axis. How can this vector be represented in terms of $\mathbf{i}$ and $\mathbf{j}$?