Cartesian coordinate system
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axis, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in threedimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real nspace) specify the point in an ndimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x^{2} + y^{2} = 4.
 $d={\sqrt {(x_{2}x_{1})^{2}+(y_{2}y_{1})^{2}}}.$
 $d={\sqrt {(x_{2}x_{1})^{2}+(y_{2}y_{1})^{2}+(z_{2}z_{1})^{2}}},$
 $(x',y')=(x+a,y+b).$
 $x'=x\cos \theta y\sin \theta$
 $y'=x\sin \theta +y\cos \theta .$
 $x'=x\cos 2\theta +y\sin 2\theta$
 $y'=x\sin 2\theta y\cos 2\theta .$
 $(x',y')=(x,y)A+b\,$
 $x'=xA_{11}+yA_{21}+b_{1}\,$
 $y'=xA_{12}+yA_{22}+b_{2},\,$


$A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}.$ [Note the use of row vectors for point coordinates and that the matrix is written on the right.]

$A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}.$ [Note the use of row vectors for point coordinates and that the matrix is written on the right.]
 $A_{11}A_{21}+A_{12}A_{22}=0$
 $A_{11}^{2}+A_{12}^{2}=A_{21}^{2}+A_{22}^{2}=1.$
 $A_{11}A_{22}A_{21}A_{12}=1.$
 $A_{11}A_{22}A_{21}A_{12}=1.$


${\begin{pmatrix}A_{11}&A_{21}&b_{1}\\A_{12}&A_{22}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.$ [Note the matrix A from above was transposed. The matrix is on the left and column vectors for point coordinates are used.]

${\begin{pmatrix}A_{11}&A_{21}&b_{1}\\A_{12}&A_{22}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.$ [Note the matrix A from above was transposed. The matrix is on the left and column vectors for point coordinates are used.]
 $(x',y')=(mx,my).$
 $(x',y')=(x+ys,y)\,$
 $(x',y')=(x,xs+y)\,$
 $\mathbf {r} =x\mathbf {i} +y\mathbf {j}$
 $\mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k}$

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