norm

The norm of a vector is simply its length: how far its tip sits from the origin. It is written ||v||. For the arrow (3,4), the norm is 5, the very hypotenuse the Pythagorean theorem gives you.

It comes straight from the dot product: ||v|| = sqrt(v . v) = sqrt(sum of the squares of the coordinates). You square each coordinate, add them up, and take the square root, which is Pythagoras working in any number of dimensions.

A norm is always zero or positive, and it is zero only for the zero vector. Dividing a vector by its own norm shrinks it to length 1 without changing its direction, a move called normalizing.