Euclidean Space - Euclidean Structure

Euclidean Structure

Euclidean space is more than just a real coordinate space. In order to apply Euclidean geometry one needs to be able to talk about the distances between points and the angles between lines or vectors. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on Rn. The inner product of any two real n-vectors x and y is defined by

The result is always a real number. Furthermore, the inner product of x with itself is always nonnegative. This product allows us to define the "length" of a vector x as

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn.

The (non-reflex) angle θ (0° ≤ θ ≤ 180°) between x and y is then given by

where cos−1 is the arccosine function.

Finally, one can use the norm to define a metric (or distance function) on Rn by

This distance function is called the Euclidean metric. It can be viewed as a form of the Pythagorean theorem.

Real coordinate space together with this Euclidean structure is called Euclidean space and often denoted En. (Many authors refer to Rn itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure makes En an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Rotations of Euclidean space are then defined as orientation-preserving linear transformations T that preserve angles and lengths:

In the language of matrices, rotations are special orthogonal matrices.