Unit Vector
Vector of length one in the same direction.
Formula
Variables
Example
Normalizes to length 1
How it works
A unit vector is created by dividing a nonzero vector by its own length, so the direction stays the same while the magnitude becomes 1. This works because scaling a vector by the reciprocal of its norm shrinks or stretches it without changing where it points. It is a standard way to separate direction from size in vector calculations.
When to use it
Use this when you need direction only, such as describing motion, building coordinate axes, or projecting one vector onto another. It is also useful in physics, computer graphics, and machine learning whenever a direction needs to be represented independently of magnitude.
Worked example, step by step
Let \mathbf{v} = \langle 3, 4 \rangle Step 1: Find the length of \mathbf{v} \|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 Step 2: Divide each component by the length \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{\langle 3, 4 \rangle}{5} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle Step 3: Check the result \left\|\hat{\mathbf{v}}\right\| = \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{1} = 1 Final answer: \hat{\mathbf{v}} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle
Common mistakes to avoid
- Dividing by the vector itself instead of by its magnitude
- Forgetting that the zero vector has no unit vector because its length is 0
- Changing the direction by using absolute values or signs incorrectly
Did You Know?
Normalising a vector to length 1 keeps only its direction, essential for representing orientations in physics and graphics.
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