3D Vector Magnitude Calculator

3D vector magnitude: |v| = √(x² + y² + z²)

The magnitude of a 3D vector v = (x, y, z) is just its Euclidean length, |v| = √(x² + y² + z²). Think of it as the Pythagorean theorem stretched into a third axis. In 2D the hypotenuse of a right triangle comes out to √(x² + y²); add a z and you get the diagonal of a rectangular box with sides x, y, z, which is √(x² + y² + z²). Take v = (2, 3, 6): |v| = √(4 + 9 + 36) = √49 = 7. Divide each component by |v| and you have the unit vector v̂ = v / |v|. Its length is exactly 1, and it points the same way v did, which is why graphics code leans on it for shading normals and physics keeps direction separate from intensity that way. The same magnitude shows up in the dot product u·v = |u||v|cos θ and the cross product |u×v| = |u||v|sin θ.

Applications

It turns up wherever a length in space matters. CAD/CAM and 3D printing use it for the distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂). Physics uses it for the magnitude of velocity, force or acceleration vectors. GPS and aerospace work in ECEF coordinates, which are 3D vectors measured from the Earth's centre. Bioinformatics measures the distance between atoms in a protein or the RMSD between structures, and computer graphics normalises vectors for lighting, shading and physics simulations.

FAQ

Why use 3D instead of 2D? Because the real world has three dimensions. You can't represent altitude in navigation, depth in a graphics scene, or a z-coordinate in physics without that third axis.

How do I get the unit vector? Divide each component by the magnitude, so v̂ = (x/|v|, y/|v|, z/|v|). What comes back has length 1 and still points the way v did.

Does it work for vectors with negative components? Yes. Each component gets squared, which wipes out the sign. Magnitude cares about how big each coordinate is, never whether it's positive or negative.