Magnitude (mathematics) - Euclidean Vectors

Euclidean Vectors

A Euclidean vector represents the position of a point P in a Euclidean space. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P): x = . Its magnitude or length is most commonly defined as its Euclidean norm (or Euclidean length):

For instance, in a 3-dimensional space, the magnitude of is √(42 + 52 + 62) = √77 or about 8.775. This is equivalent to the square root of the dot product of the vector by itself:

The Euclidean norm of a vector is just a special case of Euclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:

The second notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinants of matrices.