Unit Vectors

Video lesson · from 28:361 worked example

One lesson video covers all of Vectors, so it opens at 28:36 for this topic — not from the beginning.

Any vector with a magnitude of 1 is called a unit vector.

e.g. Let $\vec{u} = \begin{pmatrix} \sqrt{3}/2 \\ 0 \\ 1/2 \end{pmatrix}$ , then

|\vec{u}| = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + 0^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + 0 + \frac{1}{4}} = \sqrt{1} = 1

Therefore $\vec{u}$ is a unit vector.

To find a unit vector parallel to vector $\vec{v}$ , we divide $\vec{v}$ by its magnitude: $\frac{1}{|\vec{v}|}\vec{v}$ .

Divide by the magnitude: a unit vector is $\frac{1}{|\vec{u}|}\vec{u}$ . Dividing by a single component instead is a common error.
Its magnitude must be 1: that is the definition, and squaring your components and adding is a quick check.
Keep the surd: the magnitude is usually irrational — leave it exact rather than rounding, or the result will not have magnitude 1.
Direction is unchanged: a unit vector points the same way as the original; only its length changes.

Example 1

Find the components of the unit vector $\vec{u}$ parallel to vector $\vec{v}$ if $\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .

First find the magnitude of $\vec{v}$ :

|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

The unit vector $\vec{u}$ is:

\vec{u} = \frac{1}{5}\vec{v} = \frac{1}{5}\begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 3/5 \\ 4/5 \end{pmatrix}