# Magnitude

The magnitude of a vector v is denoted ||v|| and represents the absolute-value length of v.

Note: Another name for the magnitude of v is the Euclidean norm of v, in honor of Euclid, one of the first mathematicians to do serious work concerning the geometry of length, distance, and angles.

For example, the length of v = [5,7]^{T} is given by the Pythagorean theorem:

The length of a 3-vector such as v = [2, 4, -3]^{T} is also given by the Pythagorean theorem:

In general, if v is the n-vector,

then:

### Effect of scalar multiplication

If v is a vector and c is a real-number scalar, then the magnitude of the scaled vector cv is given by:

This should make sense because if we stretch v by a factor of c, then the length of v should be stretched by a factor of |c| since ||cv|| measures the absolute-value length of vector cv.

See also vectors.