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1. Cartesian Coordinates

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   Cartesian coordinates, also known as rectangular coordinates, constitute a coordinate system introduced by René Descartes in the 17th century, formalizing the relationship between geometry and algebra. This system defines the position of a point in $n$-dimensional Euclidean space by $n$ ordered, signed distances from a set of mutually orthogonal reference lines, called axes. In the familiar two-dimensional plane ($\mathbb{R}^2$), a point $P$ is unique…

2. Cartesian Coordinates

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   Transformation and Coordinate Invariance
   A key feature of the Cartesian system is its rigidity; transformations between different Cartesian systems sharing the same origin are restricted to rotations. Rotations are governed by orthogonal matrices ($R$), such that $R R^T = I$.
   If a system $(x, y, z)$ is rotated to a new system $(x', y', z')$, the relationship is expressed as:

3. Spatial Coordinate

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   Cartesian Coordinates
   The Cartesian system, sometimes known as the rectangular coordinate system, defines a point $P$ by its perpendicular distances from a set of mutually orthogonal axes, designated $X$, $Y$, and $Z$. The coordinates are represented as an ordered triplet $(x, y, z)$.
   The relationship between two points $P1(x1, y1, z1)$ and $P2(x2, y2, z2)$ in Euclidean space is governed by the distance formula: