#### Answer

Let $\mathbf{v}$ be a nonzero vector. If $\theta $ is the direction angle measured from the positive x-axis to v, then the vector can be expressed in terms of its magnitude and direction angle as:
$\mathbf{v}=\left\| \mathbf{v} \right\|\text{ }\underline{\cos \theta }\text{ }\mathbf{i}+\left\| \mathbf{v} \right\|\text{ }\underline{\sin \theta }\text{ }\mathbf{j}$

#### Work Step by Step

For representing any vector in the two systems of vectors, $\mathbf{i}$, and $\mathbf{j}$, which are mutually perpendicular with each other, we make an angle $\theta $ along the positive x-axis; then vector can be expressed by calculating three quantities: magnitude, $\cos \theta $, and $\sin \theta $, and it can be expressed as below:
$\mathbf{v}=\left\| \mathbf{v} \right\|\text{ }\underline{\cos \theta }\text{ }\mathbf{i}+\left\| \mathbf{v} \right\|\text{ }\underline{\sin \theta }\text{ }\mathbf{j}$
Where $\left\| \mathbf{v} \right\|$ is the magnitude of vector $\mathbf{v}$.