Vector Magnitude Calculator

What Is Vector Magnitude?

The magnitude (or norm) of a vector is its length. For $\mathbf{v} = (x, y, z)$ in 3D:

$$|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}$$

This is the Pythagorean theorem extended to higher dimensions. Magnitude is always non-negative. Related: Unit Vector Calculator, Vector Direction Calculator, and Distance Calculator.

For v = (3, -1, 2, -3) in 4D: $|v| = \sqrt{9 + 1 + 4 + 9} = \sqrt{23} \approx 4.796$.

How do I find the magnitude of a vector?

Square each component, add the squares, then take the square root.

Can vector magnitude be negative?

No. Magnitude is a length, so it is always zero or positive.

What is the magnitude of a unit vector?

Exactly 1. Unit vectors are normalized to have length 1.

How is magnitude related to the dot product?

|v| = √(v · v). The dot product of a vector with itself equals the squared magnitude.