Unit Vector

8. Unit Vector

A vector that has a norm (length) of exactly 1.

For a vector $\vec{𝑣}$ in $𝑛$ -dimensional space, the unit vector $\hat{𝑣}$ in the same direction is:

\hat{𝑣} = \frac{\vec{𝑣}}{‖ \vec{𝑣} ‖}

Key properties:

Example

Find the unit vector with the same direction as $\vec{𝑎} = [\begin{matrix} 3 \\ 4 \end{matrix}]$ .

‖ \vec{𝑎} ‖ = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5

\hat{𝑢} = \frac{\vec{𝑎}}{‖} \vec{𝑎} ‖ = [\begin{matrix} \frac{3}{5} \\ \frac{4}{5} \end{matrix}]

‖ \hat{𝑢} ‖ = \sqrt{{(\frac{3}{5})}^{2} + {(\frac{4}{5})}^{2}} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{1} = 1 ✓