Unit Vectors

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: $$ \begin{gather} \hat{\smash{\imath}\vphantom{i}} = \langle 1,0\rangle\cr \hat{\smash{\jmath}\vphantom{j}} = \langle 0,1\rangle \end{gather} $$

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
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


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