# What Does Unit Vector Mean?

Unit vectors are vectors with a length of $1$. All vectors have a corresponding unit vector. You can find it by using this formula:

Formula

### UnitVector

You also have unit vectors that sit along the axes. All vectors in the plane are made up of a combination of these.

Theory

### TheUnitVectorsAlongtheCoordinateAxes

• The unit vector along the $x$-axis: ${\stackrel{\to }{e}}_{x}=\phantom{\rule{-0.17em}{0ex}}\left(1,0\right)$.

• The unit vector along the $y$-axis: ${\stackrel{\to }{e}}_{y}=\phantom{\rule{-0.17em}{0ex}}\left(0,1\right)$.

A vector can always be expressed using the unit vectors ${\stackrel{\to }{e}}_{x}$ and ${\stackrel{\to }{e}}_{y}$ like this:

 $\phantom{\rule{-0.17em}{0ex}}\left(a,b\right)=a{\stackrel{\to }{e}}_{x}+b{\stackrel{\to }{e}}_{y}$

Example 1

Find the unit vector of $\phantom{\rule{-0.17em}{0ex}}\left(3,4\right)$.

Example 2

Write the vector $\phantom{\rule{-0.17em}{0ex}}\left(-3,5\right)$ by using the unit vectors that sit along the two axes.

You know that the unit vectors of the axes are ${\stackrel{\to }{e}}_{x}$ and ${\stackrel{\to }{e}}_{y}$, and that those are ${\stackrel{\to }{e}}_{x}=\phantom{\rule{-0.17em}{0ex}}\left(1,0\right)$ and ${\stackrel{\to }{e}}_{y}=\phantom{\rule{-0.17em}{0ex}}\left(0,1\right)$. That means you can decompose the vector and find the expression you’re looking for. It will look like this:

$\begin{array}{llll}\hfill \phantom{\rule{-0.17em}{0ex}}\left(-3,5\right)& =\phantom{\rule{-0.17em}{0ex}}\left(-3,0\right)+\phantom{\rule{-0.17em}{0ex}}\left(0,5\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-3\phantom{\rule{-0.17em}{0ex}}\left(1,0\right)+5\phantom{\rule{-0.17em}{0ex}}\left(0,1\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-3\cdot {\stackrel{\to }{e}}_{x}+5\cdot {\stackrel{\to }{e}}_{y}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$