Finding a Unit Vector

The formula

A unit vector has magnitude 1. To find the unit vector pointing the same way as any vector $\vec{a}$, divide by its own magnitude:

Worked example

Find the unit vector in the direction of $\vec{a} = \hat{i}+\hat{j}+2\hat{k}$.

Solution: $|\vec{a}| = \sqrt{1^2+1^2+2^2} = \sqrt{6}$, so:

Worked example: between two points

Find the unit vector in the direction of $\overrightarrow{PQ}$ where $P(1,2,3)$ and $Q(4,5,6)$.

Solution: $\overrightarrow{PQ}$ is found by subtracting coordinates: $(4-1, 5-2, 6-3) = (3,3,3) = 3\hat{i}+3\hat{j}+3\hat{k}$. Its magnitude is $\sqrt{27} = 3\sqrt{3}$, so:

Worked example: scaling to a target magnitude

Find a vector in the direction of $\vec{a} = \hat{i}-2\hat{j}$ with magnitude 7.

Solution: First find the unit vector, then scale it up to magnitude 7. $|\vec{a}| = \sqrt{1+4} = \sqrt{5}$: