# Unit Vectors

• PRACTICE (online exercises and printable worksheets)
• You may want to review prior sections:

A unit vector is a vector that has length $\,1\,$.

When working with vectors $\,\langle a,b\rangle\,$,
two unit vectors are singled out as being particularly important, and are given special names: $$\begin{gather} \hat{\smash{\imath}\vphantom{i}} = \langle 1,0\rangle\cr \hat{\smash{\jmath}\vphantom{j}} = \langle 0,1\rangle \end{gather}$$

Read $\,\hat{\smash{\imath}\vphantom{i}}\,$ aloud as ‘ i hat ’.
Read $\,\hat{\smash{\jmath}\vphantom{j}}\,$ aloud as ‘ j hat ’.
• Drop the Dots
When you put a ‘hat’ on ‘ i ’ or ‘ j ’, you drop the dot:
• write $\,\hat{\smash{\imath}\vphantom{i}}\,$, NOT $\,\hat{i}\,$
• write $\,\hat{\smash{\jmath}\vphantom{j}}\,$, NOT $\,\hat{j}\,$
• Another Name for Vectors—Unit Vector Form
Every vector $\,\langle a,b\rangle\,$ can be easily expressed in terms of $\,\hat{\smash{\imath}\vphantom{i}}\,$ and $\,\hat{\smash{\jmath}\vphantom{j}}\,$, as follows: \begin{align} \langle a,b\rangle \quad &= \quad \langle a,0\rangle + \langle 0,b\rangle\cr &= \quad a\langle 1,0\rangle + b\langle 0,1\rangle\cr &= \quad a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}} \end{align} The name ‘$\,a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}}\,$’ is called the unit vector form of the vector $\,\langle a,b\rangle\,$.
• There are infinitely many unit vectors
However, $\,\hat{\smash{\imath}\vphantom{i}}\,$ and $\,\hat{\smash{\jmath}\vphantom{j}}\,$ are simplest.
Any vector from the center
to a point on this circle has length $\,1\,$,
and hence is a unit vector.

## Adding Vectors in Unit Vector Form

\begin{align} \color{red}{(a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}}) + (c\hat{\smash{\imath}\vphantom{i}} + d\hat{\smash{\jmath}\vphantom{j}})} \quad &= \quad \langle a,b\rangle + \langle c,d\rangle\cr &= \quad \langle a+c\,,\,b+d\rangle\cr &= \quad \color{red}{(a+c)\hat{\smash{\imath}\vphantom{i}} + (b+d)\hat{\smash{\jmath}\vphantom{j}}} \end{align}

## Multiplying Vectors in Unit Vector Form by a Constant

\begin{align} \color{red}{k(a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}})} \quad &= \quad k\langle a,b\rangle\cr &= \quad \langle ka\,,\,kb\rangle\cr &= \quad \color{red}{ka\hat{\smash{\imath}\vphantom{i}} + kb\hat{\smash{\jmath}\vphantom{j}}} \end{align}

The idea is as simple as ‘combining like terms’!
Just gather together the $\,\hat{\smash{\imath}\vphantom{i}}\,$ and $\,\hat{\smash{\jmath}\vphantom{j}}\,$ terms separately.

You don't always have $\,\hat{\smash{\imath}\vphantom{i}}\,$ first and $\,\hat{\smash{\jmath}\vphantom{j}}\,$ second, so be careful.
They're often all mixed up.

Here's an example:
\begin{align} 3\hat{\smash{\jmath}\vphantom{j}} + 7\hat{\smash{\imath}\vphantom{i}} - 5(2\hat{\smash{\imath}\vphantom{i}} - \hat{\smash{\jmath}\vphantom{ij}}) \quad &=\quad 3\hat{\smash{\jmath}\vphantom{j}} + 7\hat{\smash{\imath}\vphantom{i}} - 10\hat{\smash{\imath}\vphantom{i}} + 5\hat{\smash{\jmath}\vphantom{ij}}\cr\cr \ &=\quad -3\hat{\smash{\imath}\vphantom{i}} + 8\hat{\smash{\jmath}\vphantom{j}} \end{align}

Master the ideas from this section