Math Wiki
Advertisement

Three-dimensional space (Also known specifically as 3-space or tri-dimensional space) is a geometrical setting in which three individual values (also called parameters) are obligatory to determine the position of a mathematical element (I.e Point, Ray and etc.). This is the informal meaning of the word (or term) "dimension".

In Physics and Mathematics, a sequence of repeating n numbers can be implied as a location in n-dimensional space. When the equation n = 3, the mathematical set of all selected locations is called Three-dimensional Euclidean space. This dimensional number can be characterized as the equation R^3. This reveals that it can serve as the parameter model of the entire physical universe (which includes the spatial part, without considering the consonant, time) in which all types of matter exist. However, this space is known to only be a single example of a extensive variety of spaces in this dimension called 3-manifolds. One of the classical examples when the three explicit values refer to idiosyncratic measurements in contrasting directions (coordinates), any three directions in the space can be chosen, which provides that the vectors in these directions do not be active in the same 2-space (plane version). Furthermore in these cases, these three values can be labled by any kind of combination that is chosen by three from the specific five terms: widthheightdepth, and length.


In Euclidean Geometry

Coordinate System

In Mathematics and Analytic Geometry (also known as or called Cartseian Geometry) describes every point in three-dimensional space by the specific means of the three coordinates. Three of these are given, with each of them being perpendicular to the alternative two at the origin, the point at which they caville. These are usually represented by the last three letters of the alphabet or, xy, and z. Relative to these three axes, the located position of any point in the three-dimensional space is given by a ordered tripled list of these real numbers, with each of the number giving the distance of that point from the origin that is measured along the given part of the axis. The axis is known to be equal to the specific distance of that point from the locationally determined point by the other two axes.

Other specific popular methods in Euclidean Geometry of describing the positional location of the point in three-dimensional space which include cylindrical coordinates and spherical coordinates, though there is known to be a infinite sequence of possible methods.

Lines and Planes

Two of these distinct points can always determine a (straight) line. Four of these distinct points can have the ability to either be collinear, coplanar, or determine the entire dimensional space.

Two of these lines can also have the ability to intersect, be parallel or skew. Two parallel or intersecting lines have the ability to lie in a unique plane. So, the skew lines in geometry can be lines that are lines that deceit and do not lie in a very common place.

Two distinct planes either have the ability to meet in a universal line or can be parallel (which means, do not meet). Three of these distinct, no specific pair of which are known to be parallel, which can either meet eachother in a common line. They can also meet in a unique common point or can have no point in common. In the last specific case, there are specific lines in the plane that are parallel given to the line.

A hyperplane is known to be a subspace of one specific dimension less than the dimension of it's full space. The Hyperplanes of a three-dimensional space are resulted as the two-dimensional subspaces, which that is, the planes. In specific terms of Cartesian coordinates, the specific points of the hypersphere satisfy a single linear mathematical statement. So the planes in the 3-space are described by it. A line can be described by a distinctive pair of independent linear equations, with each of them representing a plane having this line as a accepted intersection.

Spheres and Balls

A sphere in 3-dimensional space (3-sphere, or a 2-sphere due to it being a Two-dimensional sphere or object) consists of the all-set points in 3-space at a fixed distance of r from a central point called P. The enclosed stable part by the sphere is simply called a ball (or precisely, a 3-ball). The volume of the 3-ball is given by:

Failed to parse (syntax error): {\displaystyle V=4/3πr^3}

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

Polytopes

Main Article: Polyhedron

Surfaces of Revolution

A type of surface that is generated by revolving a plane curved with a fixed line in it's place as an axis is known as the surface of revolution. The part of it (plane) is called a generatix. A Section of the surface, which is made by intersecting the with a distinctive plane that is orthogonal (perpendicular) to the axis, is a geometrical circle.

Some examples occour when the plane is a line. If the plane line intersects the axis line, the revoultion is a right circular cone with the apex (also known as the vertex), the point of the intersection. However, if both Generatix and axis are parallel, the surface of a revolution is a circular cylinder.

Quadric Surfaces

In analogy with its conic sections, the sets of points whose cartseian coordinates satisfy the accoustomed equation of the second degree. This namely means, 

Where the letters, A,B,C,F,G,H,J,K,L and M are known to be real numbers and not all of these letters are real. These numbers are known to include A,B,C,F,G, and H are not real but instead called a quadratic surface.

There are known to be six categories of non-degenerate quadric surfaces:

  1. Ellipsoid
  2. Hyperboloid of one sheet
  3. Hyperboloid of two sheets
  4. Elliptic Cone
  5. Elliptic Parabololid
  6. Hyperbolic Paraboloid

The degenerate quadratic surfaces are known to be the empty set, a single point, line, plane, a pair of planes with a quadratic cylinder (a surface that is consisting of a non-degenerate conic section in the π plane and all the lines of R^3 through that the conic that are normal to π). Elliptic cones can be sometimes considered to be generate expanses as well.

Both the hyperboloid of the first sheet and the hyperbolic of the paraboloid are ruled areas, meaning that these two are made up from a group of straight lines. In fact, each has two different families of generating lines, the known members of each family are known to be disjoint and each group member intersects with only just one of a exception, every other member of the family. With each part of the group being a regulus.

In Linear Algebra

Another way of viewing the three-dimensional space is discovered and found in Linear Algebra, where the idea of its independence is known to be crucial. The space is known to have three dimensions due to the length of a single box is autonomous of the width or breadth. In Linear Algebra's technical language, space is three-dimensional because every point in space can be described as a linear combination of three basis vectors.

Dot product, angle, and length


Main Article: Dot product

A single vector can be imaginated as an arrow. The vector's magnitude is it's length, and it's direction is the specific direction the arrow is pointing at. A vector in R^3 can be represented by a combined order of tripled numbers. These numbers are acknowledged to be called the components of the vector.

The dot product of two factors A = [A1,A2,A3] and B = [B1,B2,B3] can be (is) defined as:

A x B = A1 and B1 + A2 and B2 + A3 and B3.

The magnitude of the vector, A, is defined as //A//

Advertisement