Unit Vector Calculator

What Is a Unit Vector?

A unit vector points in the same direction as the original vector but has a magnitude (length) of exactly 1. It is written as û and computed by dividing each component by the vector's magnitude.

Unit Vector Formula

The unit vector formula is:

$$\hat{u} = \frac{u}{|u|}$$

where $|u| = \sqrt{x^2 + y^2 + z^2}$ for a 3D vector.

Example

For u = (8, -3, 5), the magnitude is $\sqrt{98} \approx 9.899$. The unit vector is $(8/9.899,\ -3/9.899,\ 5/9.899) \approx (0.808,\ -0.303,\ 0.505)$.

Frequently Asked Questions

How do I find a unit vector in the same direction?

Divide each component of the vector by its magnitude. The result has length 1 and points the same way.

Is (1, 1) a unit vector?

No. Its magnitude is $\sqrt{2} \approx 1.414$, not 1. The unit vector in that direction is $(1/\sqrt{2},\ 1/\sqrt{2})$.

What is the magnitude of a unit vector?

Always 1. By definition, unit vectors have length exactly 1.

Can I normalize a zero vector?

No. The zero vector (0, 0) or (0, 0, 0) has magnitude 0, so division is undefined. Direction is not defined for the zero vector.

Report

What Is a Unit Vector?

Unit Vector Formula

Example

Frequently Asked Questions

Related Tools

Recent Visits