Four-dimensional Space - Vectors

Vectors

Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example a general point might have position vector a, equal to

This can be written in terms of the four standard basis vectors (e₁, e₂, e₃, e₄), given by

so the general vector a is

Vectors add, subtract and scale as in three dimensions.

The dot product of Euclidean three-dimensional space generalizes to four dimensions as

It can be used to calculate the norm or length of a vector,

and calculate or define the angle between two vectors as

Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.

The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e₁₂, e₁₃, e₁₄, e₂₃, e₂₄, e₃₄). They can be used to generate rotations in four dimensions.