Three-dimensional space (mathematics)

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Three-dimensional Cartesian coordinate system with the x-axis pointing towards the observer.
(See diagram description for correction.)

Three-dimensional space (also: tri-dimensional space) is a geometric three-parameter model of the physical universe (without considering time) in which all known matter exists. These three dimensions can be labeled by a combination of three chosen from the terms length, width, height, depth, and breadth. Any three directions can be chosen, provided that they do not all lie in the same plane.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol \scriptstyle{\mathbb{R}}^3. This space is only one example of a great variety of spaces in three dimensions called 3-manifolds.

In geometry[edit]

Coordinate systems[edit]

Main article: Coordinate system


Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there is an infinite number of possible methods. See Euclidean space.

Below are images of the above-mentioned systems.


Main article: Polyhedron


Regular polytopes in three dimensions
Class Platonic solids Kepler-Poinsot polyhedra
Symmetry Td Oh Ih
Order 24 48 120


Main article: Sphere
A two-dimensional perspective projection of a sphere

A sphere in 3-space (also called a 2-sphere because its surface is 2-dimensional) consists of the set of all points in 3-space at a fixed distance r from a central point P. The volume enclosed by this surface is:


Another type of sphere, but having a three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space \mathbb{R}^4 at distance one. If any position is P=(x,y,z,t), then x^2+y^2+z^2+t^2=1 characterize a point in the 3-sphere.


In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: north/south (latitude), east/west (longitude) and up/down (altitude). These pairs of directions are mutually orthogonal: They are at right angles to each other. Movement along one axis does not change the coordinate value of the other two axes. In mathematical terms, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.

In linear algebra[edit]


Dot product, angle, and length[edit]

Main article: Dot product


\mathbf{A}\cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3

A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by \|\mathbf{A}\|. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by[2]

\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,

where θ is the angle between A and B.

The dot product of a vector A by itself is

\mathbf A\cdot\mathbf A = \|\mathbf A\|^2,

which gives

 \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},

the formula for the Euclidean length of the vector.

Cross product[edit]

Main article: Cross product





In calculus[edit]

Gradient, divergence and curl[edit]

In a rectangular coordinate system, the gradient is given by

\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y}  \mathbf{j} +
\frac{\partial f}{\partial z} \mathbf{k}


+\frac{\partial W}{\partial z


{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 \\  F_x & F_y & F_z \end{vmatrix}


\left(\frac{\partial F_z}{\partial y}  - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}

Line integrals, surface integrals, and volume integrals[edit]


\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.



\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.



\iint_{S} f \,\mathrm dS
= \iint_{T} f(\mathbf{x}(s, t)) \left\|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right\| \mathrm ds\, \mathrm dt



It can also mean a triple integral within a region D in R3 of a function f(x,y,z), and is usually written as:

\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.

Fundamental theorem of line integrals[edit]


Let  \varphi : U \subseteq \mathbb{R}^n \to \mathbb{R}. Then

 \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\varphi(\mathbf{r})\cdot d\mathbf{r}.

Stokes' theorem[edit]

Main article: Stokes' theorem



Divergence theorem[edit]

Main article: Divergence theorem

Suppose V is a subset of \mathbb{R}^n (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with V = S). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:[5]

\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV=\oiint\scriptstyle S(\mathbf{F}\cdot\mathbf{n})\,dS .


In topology[edit]


With the space \scriptstyle{\mathbb{R}}^3, the topologists locally model all other 3-manifolds.

See also[edit]


  1. ^ S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1. 
  2. ^ M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7. 
  3. ^ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space. 
  4. ^ Arfken, p. 43.

External links[edit]