structural:scholar_edition
Abstract description

It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in a subtle way. By definition, there exists a basis B = { e 1, e 2, e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V}. This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}}: the construction for the isomorphism is found here. However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V}.

On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}}, which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} }, that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }. This allows the definition of canonical projections, π i: R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} }, where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3}. For example, π 1 (x 1, x 2, x 3) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x}. This then allows the definition of the standard basis B Standard = { E 1, E 2, E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by

where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. Written out in full, the standard basis is

Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.

As opposed to a general vector space V {\displaystyle V}, the space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space.

Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space.

Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations.