Unit Vectors

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• How to Read Aloud
  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: $$ \begin{align} \cssId{s17}{\langle a,b\rangle} \quad &\cssId{s18}{= \quad \langle a,0\rangle + \langle 0,b\rangle}\cr &\cssId{s19}{= \quad a\langle 1,0\rangle + b\langle 0,1\rangle}\cr &\cssId{s20}{= \quad a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}}} \end{align} $$ 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

$$ \begin{align} \cssId{s26}{ \color{red}{(a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}}) + (c\hat{\smash{\imath}\vphantom{i}} + d\hat{\smash{\jmath}\vphantom{j}})}} \quad &\cssId{s27}{= \quad \langle a,b\rangle + \langle c,d\rangle}\cr &\cssId{s28}{= \quad \langle a+c\,,\,b+d\rangle}\cr &\cssId{s29}{= \quad \color{red}{(a+c)\hat{\smash{\imath}\vphantom{i}} + (b+d)\hat{\smash{\jmath}\vphantom{j}}}} \end{align} $$

Multiplying Vectors in Unit Vector Form by a Constant

$$ \begin{align} \cssId{s31}{\color{red}{k(a\hat{\smash{\imath}\vphantom{i}} + b\hat{\smash{\jmath}\vphantom{j}})}} \quad &\cssId{s32}{= \quad k\langle a,b\rangle}\cr &\cssId{s33}{= \quad \langle ka\,,\,kb\rangle}\cr &\cssId{s34}{= \quad \color{red}{ka\hat{\smash{\imath}\vphantom{i}} + kb\hat{\smash{\jmath}\vphantom{j}}}} \end{align} $$

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:
$$ \begin{align} \cssId{s40}{3\hat{\smash{\jmath}\vphantom{j}} + 7\hat{\smash{\imath}\vphantom{i}} - 5(2\hat{\smash{\imath}\vphantom{i}} - \hat{\smash{\jmath}\vphantom{ij}})} \quad &\cssId{s41}{=\quad 3\hat{\smash{\jmath}\vphantom{j}} + 7\hat{\smash{\imath}\vphantom{i}} - 10\hat{\smash{\imath}\vphantom{i}} + 5\hat{\smash{\jmath}\vphantom{ij}}}\cr\cr \ &\cssId{s42}{=\quad -3\hat{\smash{\imath}\vphantom{i}} + 8\hat{\smash{\jmath}\vphantom{j}}} \end{align} $$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Formula for the Length of a Vector