Ingeometry, athree-dimensional space (3D space,3-space or, rarely,tri-dimensional space) is amathematical space in which three values (coordinates) are required to determine theposition of apoint. Most commonly, it is thethree-dimensional Euclidean space, that is, theEuclidean space ofdimension three, which modelsphysical space. More general three-dimensional spaces are called3-manifolds. The term may also refer colloquially to a subset of space, athree-dimensional region (or 3Ddomain),[1] asolid figure.
Technically, atuple ofnnumbers can be understood as theCartesian coordinates of a location in an-dimensional Euclidean space. The set of thesen-tuples is commonly denoted and can be identified to the pair formed by an-dimensional Euclidean space and aCartesian coordinate system.Whenn = 3, this space is called thethree-dimensional Euclidean space (or simply "Euclidean space" when the context is clear).[2] Inclassical physics, it serves as a model of the physicaluniverse, in which all knownmatter exists. Whenrelativity theory is considered, it can be considered a local subspace ofspace-time.[3] While this space remains the most compelling and useful way to model the world as it is experienced,[4] it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the sameplane. Furthermore, if these directions are pairwiseperpendicular, the three values are often labeled by the termswidth/breadth,height/depth, andlength.
Books XI to XIII ofEuclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regularPlatonic solids in a sphere.
In the 17th century, three-dimensional space was described withCartesian coordinates, with the advent ofanalytic geometry developed byRené Descartes in his workLa Géométrie andPierre de Fermat in the manuscriptAd locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.
In the 19th century, developments of the geometry of three-dimensional space came withWilliam Rowan Hamilton's development of thequaternions. In fact, it was Hamilton who coined the termsscalar andvector, and they were first defined withinhis geometric framework for quaternions. Three dimensional space could then be described by quaternions which had vanishing scalar component, that is,. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements, as well as thedot product andcross product, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.
It was not untilJosiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbookVector Analysis written byEdwin Bidwell Wilson based on Gibbs' lectures.
Also during the 19th century came developments in the abstract formalism of vector spaces, with the work ofHermann Grassmann andGiuseppe Peano, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.
In mathematics,analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Threecoordinate axes are given, each perpendicular to the other two at theorigin, the point at which they cross. They are usually labeledx,y, andz. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple ofreal numbers, each number giving the distance of that point from theorigin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.[5]
Two distinct points always determine a (straight)line. Three distinct points are eithercollinear or determine a uniqueplane. On the other hand, four distinct points can either be collinear,coplanar, or determine the entire space.
Two distinct lines can either intersect, beparallel or beskew. Two parallel lines, ortwo 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.
Ahyperplane 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 singlelinear 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.
Asphere in 3-space (also called a2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distancer from a central pointP. The solid enclosed by the sphere is called aball (or, more precisely a3-ball).
The volume of the ball is given by
and the surface area of the sphere isAnother type of sphere arises from a 4-ball, whose three-dimensional surface is the3-sphere: points equidistant to the origin of the euclidean spaceR4. If a point has coordinates,P(x,y,z,w), thenx2 +y2 +z2 +w2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.
Asurface generated by revolving a planecurve about a fixed line in its plane as an axis is called asurface of revolution. The plane curve is called thegeneratrix 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 circularcone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circularcylinder.
In analogy with theconic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,whereA,B,C,F,G,H,J,K,L andM are real numbers and not all ofA,B,C,F,G andH are zero, is called aquadric surface.[6]
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 planeπ and all the lines ofR3 through that conic that are normal toπ).[6] Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid areruled 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.[7] Each family is called aregulus.
Another way of viewing three-dimensional space is found inlinear algebra, where the idea of independence is crucial. Space has three dimensions because the length of abox 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 independentvectors.
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 can be represented by an ordered triple of real numbers. These numbers are called thecomponents of the vector.
The dot product of two vectorsA = [A1,A2,A3] andB = [B1,B2,B3] is defined as:[8]
The magnitude of a vectorA is denoted by||A||. The dot product of a vectorA = [A1,A2,A3] with itself is
Thecross product orvector product is abinary operation on twovectors in three-dimensionalspace and is denoted by the symbol ×. The cross productA ×B of the vectorsA andB is a vector that isperpendicular to both and thereforenormal to the plane containing them. It has many applications in mathematics,physics, andengineering.
In function language, the cross product is a function.
The components of the cross product are, and can also be written in components, using Einstein summation convention as where is theLevi-Civita symbol. It has the property that.
Its magnitude is related to the angle between and by the identity
The space and product form analgebra over a field, which is notcommutative norassociative, but is aLie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, isisomorphic to the Lie algebra of three-dimensional rotations, denoted. In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies theJacobi identity. For any three vectors and
One can inn dimensions take the product ofn − 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 andseven dimensions.[10]
The cross-product in respect to a right-handed coordinate system
It can be useful to describe three-dimensional space as a three-dimensional vector space over the real numbers. This differs from in a subtle way. By definition, there exists a basis for. This corresponds to anisomorphism between and: the construction for the isomorphism is foundhere. However, there is no 'preferred' or 'canonical basis' for.
On the other hand, there is a preferred basis for, which is due to its description as aCartesian product of copies of, that is,. This allows the definition of canonical projections,, where. For example,. This then allows the definition of thestandard basis defined bywhere is theKronecker delta. Written out in full, the standard basis is
Therefore can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, can be obtained by starting with and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.
As opposed to a general vector space, the space is sometimes referred to as a coordinate space.[11]
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 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 in order to do concrete computations.
A more abstract description still is to model physical space as a three-dimensional affine space over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes calledEuclidean affine spaces for distinguishing them from Euclidean vector spaces.[12]
This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.
The above discussion does not involve thedot product. The dot product is an example of aninner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality). For any inner product, there exist bases under which the inner product agrees with the dot product, but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotationsSO(3).
where the expression between bars on the right-hand side is themagnitude of thecross product of thepartial derivatives ofx(s,t), and is known as the surfaceelement. Given a vector fieldv onS, that is a function that assigns to eachx inS a vectorv(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.
SupposeV is a subset of (in the case ofn = 3,V represents a volume in 3D space) which iscompact and has a piecewisesmooth boundaryS (also indicated with∂V =S). IfF is a continuously differentiable vector field defined on a neighborhood ofV, then thedivergence theorem says:[15]
The left side is avolume integral over the volumeV, the right side is thesurface integral over the boundary of the volumeV. The closed manifold∂V is quite generally the boundary ofV oriented by outward-pointingnormals, andn is the outward pointing unit normal field of the boundary∂V. (dS may be used as a shorthand forndS.)
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 aknot in a piece of string.[16]
^Massey, WS (1983). "Cross products of vectors in higher dimensional Euclidean spaces".The American Mathematical Monthly.90 (10):697–701.doi:10.2307/2323537.JSTOR2323537.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.
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