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**Three-dimensional space** (also: **3-space** or, rarely, **tri-dimensional space**) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension.

- In Euclidean geometry
- Coordinate systems
- Lines and planes
- Spheres and balls
- Polytopes
- Surfaces of revolution
- Quadric surfaces
- In linear algebra
- Dot product, angle, and length
- Cross product
- In calculus
- Gradient, divergence and curl
- Line integrals, surface integrals, and volume integrals
- Fundamental theorem of line integrals
- Stokes' theorem
- Divergence theorem
- In topology
- In finite geometry
- See also
- Notes
- References
- External links

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 ** (or simply Euclidean space when the context is clear). It is commonly represented by the symbol ℝ^{3}.^{ [1] }^{ [2] } This serves as a three-parameter model of the physical universe (that is, the spatial part, without considering time), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced,^{ [3] } it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms * width *, * height *, *depth*, and * length *.

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled *x*, *y*, and *z*. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.^{ [4] }

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

Below are images of the above-mentioned systems.

Two distinct points always determine a (straight) line. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.

Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.

Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.

A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.

A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.

Varignon's theorem states that the midpoints of any quadrilateral in ℝ^{3} form a parallelogram, and hence are coplanar.

A sphere in 3-space (also called a **2-sphere** because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance *r* from a central point P. The solid enclosed by the sphere is called a **ball** (or, more precisely a **3-ball**). The volume of the ball is given by

- .

Another type of sphere arises from a 4-ball, whose three-dimensional surface is the **3-sphere**: points equidistant to the origin of the euclidean space ℝ^{4}. If a point has coordinates, *P*(*x*, *y*, *z*, *w*), then *x*^{2} + *y*^{2} + *z*^{2} + *w*^{2} = 1 characterizes those points on the unit 3-sphere centered at the origin.

In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.

Class | Platonic solids | Kepler-Poinsot polyhedra | |||||||
---|---|---|---|---|---|---|---|---|---|

Symmetry | T_{d} | O_{h} | I_{h} | ||||||

Coxeter group | A_{3}, [3,3] | B_{3}, [4,3] | H_{3}, [5,3] | ||||||

Order | 24 | 48 | 120 | ||||||

Regular polyhedron | {3,3} | {4,3} | {3,4} | {5,3} | {3,5} | {5/2,5} | {5,5/2} | {5/2,3} | {3,5/2} |

A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the * generatrix * of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.

Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.

In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,

where *A*, *B*, *C*, *F*, *G*, *H*, *J*, *K*, *L* and *M* are real numbers and not all of *A*, *B*, *C*, *F*, *G* and *H* are zero, is called a **quadric surface**.^{ [5] }

There are six types of non-degenerate quadric surfaces:

- Ellipsoid
- Hyperboloid of one sheet
- Hyperboloid of two sheets
- Elliptic cone
- Elliptic paraboloid
- Hyperbolic paraboloid

The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of ℝ^{3} through that conic that are normal to π).^{ [5] } Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.

Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family.^{ [6] } Each family is called a regulus.

Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors.

A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in ℝ^{3} can be represented by an ordered triple of real numbers. These numbers are called the **components** of the vector.

The dot product of two vectors **A** = [*A*_{1}, *A*_{2}, *A*_{3}] and **B** = [*B*_{1}, *B*_{2}, *B*_{3}] is defined as:^{ [7] }

The magnitude of a vector **A** is denoted by ||**A**||. The dot product of a vector **A** = [*A*_{1}, *A*_{2}, *A*_{3}] with itself is

which gives

the formula for the Euclidean length of the vector.

Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors **A** and **B** is given by^{ [8] }

where *θ* is the angle between **A** and **B**.

The cross product or **vector product** is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product **a** × **b** of the vectors **a** and **b** is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.

The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

One can in *n* dimensions take the product of *n* − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.^{ [9] }

In a rectangular coordinate system, the gradient is given by

The divergence of a continuously differentiable vector field **F** = *U***i** + *V***j** + *W***k** is equal to the scalar-valued function:

Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × **F** is, for **F** composed of [*F*_{x}, *F*_{y}, *F*_{z}]:

where **i**, **j**, and **k** are the unit vectors for the *x*-, *y*-, and *z*-axes, respectively. This expands as follows:^{ [10] }

For some scalar field *f* : *U* ⊆ **R**^{n} → **R**, the line integral along a piecewise smooth curve *C* ⊂ *U* is defined as

where **r**: [a, b] → *C* is an arbitrary bijective parametrization of the curve *C* such that **r**(*a*) and **r**(*b*) give the endpoints of *C* and .

For a vector field **F** : *U* ⊆ **R**^{n} → **R**^{n}, the line integral along a piecewise smooth curve *C* ⊂ *U*, in the direction of **r**, is defined as

where · is the dot product and **r**: [a, b] → *C* is a bijective parametrization of the curve *C* such that **r**(*a*) and **r**(*b*) give the endpoints of *C*.

A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, *S*, by considering a system of curvilinear coordinates on *S*, like the latitude and longitude on a sphere. Let such a parameterization be **x**(*s*, *t*), where (*s*, *t*) varies in some region *T* in the plane. Then, the surface integral is given by

where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of **x**(*s*, *t*), and is known as the surface element. Given a vector field **v** on *S*, that is a function that assigns to each **x** in *S* a vector **v**(**x**), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.

A volume integral refers to an integral over a 3-dimensional domain.

It can also mean a triple integral within a region *D* in **R**^{3} of a function and is usually written as:

The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let . Then

Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:

Suppose V is a subset of (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:^{ [11] }

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂*V* is quite generally the boundary of V oriented by outward-pointing normals, and **n** is the outward pointing unit normal field of the boundary ∂*V*. (*d***S** may be used as a shorthand for **n***dS*.)

Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a knot in a piece of string.^{ [12] }

In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble .

Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(*q*), there is a projective space PG(3,*q*) of three dimensions. For example, any three skew lines in PG(3,*q*) are contained in exactly one regulus.^{ [13] }

- ↑ "Compendium of Mathematical Symbols".
*Math Vault*. 2020-03-01. Retrieved 2020-08-12. - ↑ "Euclidean space - Encyclopedia of Mathematics".
*encyclopediaofmath.org*. Retrieved 2020-08-12. - ↑ "Euclidean space | geometry".
*Encyclopedia Britannica*. Retrieved 2020-08-12. - ↑ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013).
*Calculus : Single and Multivariable*(6 ed.). John wiley. ISBN 978-0470-88861-2. - 1 2 Brannan, Esplen & Gray 1999 , pp. 34–5
- ↑ Brannan, Esplen & Gray 1999 , pp. 41–2
- ↑ Anton 1994 , p. 133
- ↑ Anton 1994 , p. 131
- ↑ 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.

CS1 maint: ref=harv (link) - ↑ Arfken, p. 43.
- ↑ M. R. Spiegel; S. Lipschutz; D. Spellman (2009).
*Vector Analysis*. Schaum’s Outlines (2nd ed.). USA: McGraw Hill. ISBN 978-0-07-161545-7. - ↑ Rolfsen, Dale (1976).
*Knots and Links*. Berkeley, California: Publish or Perish. ISBN 0-914098-16-0. - ↑ Albrecht Beutelspacher & Ute Rosenbaum (1998)
*Projective Geometry*, page 72, Cambridge University Press ISBN 0-521-48277-1

In vector calculus, the **curl** is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector characterize the rotation at that point.

In vector calculus, **divergence** is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

In vector calculus, the **gradient** of a scalar-valued differentiable function *f* of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in *n-*dimensional space as the vector:

In vector calculus and differential geometry, **Stokes' theorem**, also called the **generalized Stokes theorem** or the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,

In mathematics, an ** n-sphere** is a topological space that is homeomorphic to a

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

In mathematics, a **plane** is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, in two dimensions, the **normal line** to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In the mathematical field of differential geometry, one definition of a **metric tensor** is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar *g*(*v*, *w*) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

In the mathematical fields of differential geometry and tensor calculus, **differential forms** are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In mathematics, **conformal geometry** is the study of the set of angle-preserving (conformal) transformations on a space.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the **Hodge dual** of the element. This map was introduced by W. V. D. Hodge.

**Scalar potential**, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In mathematics, a **surface integral** is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field over the surface, or a vector field. If a region R is not flat, then it is called surface as shown in the illustration.

In differential geometry, the **second fundamental form** is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

In mathematics, a **norm** is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. The set ℝ^{2} of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

- Anton, Howard (1994),
*Elementary Linear Algebra*(7th ed.), John Wiley & Sons, ISBN 978-0-471-58742-2 - Arfken, George B. and Hans J. Weber.
*Mathematical Methods For Physicists*, Academic Press; 6 edition (June 21, 2005). ISBN 978-0-12-059876-2. - Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),
*Geometry*, Cambridge University Press, ISBN 978-0-521-59787-6

Wikiquote has quotations related to: Three-dimensional space |

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*three-dimensional*at Wiktionary- Weisstein, Eric W. "Four-Dimensional Geometry".
*MathWorld*. - Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry Keith Matthews from University of Queensland, 1991

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