Unit vectors

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A unit vector is a vector with magnitude 1. The unit vector to a given vector $\vec{\mathbf{a}}$ can be determinde by dividing the given vector by its magnitude $|\vec{\mathbf{a}}|$:

$\vec{\textbf{e}}_{a} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} = \frac{\vec{\textbf{a}}}{\sqrt{a_x^2 + a_y^2 + a_z^2}} = \frac{1}{\sqrt{a_x^2 + a_y^2 + a_z^2}} \begin{bmatrix} a_x\\ a_y\\ a_z \end{bmatrix}$

The vector $\vec{\mathbf{e}}_{a}$ has the magnitude 1 (so $|\vec{\mathbf{e}}_a|=1$) and is pointed to the direction of $\vec{\mathbf{a}}$. So every vector can be described by its magnitude (so a scalar value) and the corresponding unit vector. Therefore $\vec{\mathbf{a}}$ can also be written as follows:

$\vec{\textbf{a}} = \frac{\vec{\textbf{a}}}{|\vec{\textbf{a}}|} |\vec{\textbf{a}}| = \vec{\textbf{e}}_{a} |\vec{\textbf{a}}|$