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Cartesian coordinate system

The Cartesian coordinate system, also known as the rectangular coordinate system, is a fundamental mathematical tool that uniquely identifies each point in a or using ordered numerical coordinates relative to intersecting axes, typically labeled as the x-axis (horizontal) and y-axis (vertical) in two dimensions, with an at their intersection point (0,0). This system extends naturally to three dimensions by adding a z-axis to the , dividing into octants based on the signs of the coordinates. Each coordinate represents the signed distance from the along the respective axis, enabling precise algebraic representation of geometric shapes and relations. Named after René Descartes (1596–1650), the system was formalized in his 1637 treatise , where he demonstrated how to translate geometric problems into algebraic equations by assigning coordinates to points. However, the core idea of linking algebra and geometry through coordinates was developed independently around the same time by French mathematician (1601–1665), who by 1630 had devised a method to plot quantities as points on a plane, though he did not publish it during his lifetime. Precursors to this innovation include 14th-century graphs by and algebraic-geometric connections by in the late 16th century, but Descartes' and Fermat's work marked the birth of . The Cartesian system revolutionized by reducing complex geometric constructions—previously reliant on qualitative diagrams as in methods—to solvable algebraic equations, paving the way for advancements in , physics, and . Its perpendicular axes and uniform scaling allow for straightforward calculations of distances, slopes, and curves, such as representing a line as y = mx + b or a circle as x² + y² = r². Today, it underpins virtually all graphical and computational modeling in sciences, from plotting functions in two dimensions to analysis in .

History

Origins in analytic geometry

The Cartesian coordinate system emerged as a pivotal innovation within the broader development of in the , marking a fundamental shift from the synthetic methods of ancient , exemplified by Euclid's Elements, to algebraic approaches that allowed for the manipulation of geometric forms through equations. , reliant on axioms, postulates, and constructions with and , had dominated mathematical thought for centuries but proved limited for handling complex curves and loci. This transition enabled mathematicians to represent spatial relationships numerically, transforming into a branch amenable to algebraic techniques and paving the way for the study of curves via coordinate assignments. René Descartes played a central role in this evolution through his 1637 treatise , appended to , where he introduced a "geometrical " that systematically applied algebraic notation to geometric problems. Descartes assigned coordinates—such as lengths along lines—to points on plane figures, allowing him to translate constructions into equations and solve them via algebraic operations, thereby forging a direct link between and . For instance, in addressing the classical Pappus problem involving four lines, Descartes demonstrated how such coordinate-based equations, typically of the second degree, could generate and classify conic sections like parabolas, hyperbolas, and ellipses, revealing that these shapes are unified representations of relations rather than disparate entities requiring separate synthetic proofs. Independently, developed similar ideas around 1636, as evidenced in his unpublished manuscripts, where he employed coordinates to describe loci of points satisfying algebraic conditions, further advancing the algebraic treatment of curves. Fermat's method of , a precursor to techniques, involved assuming a curve and a nearby point "adequal" (nearly equal) to derive tangents by balancing differences in algebraic expressions, thus enabling precise geometric analysis without traditional construction. This approach complemented Descartes' framework by emphasizing variational methods for curve properties, collectively solidifying analytic geometry's capacity to manipulate curves algebraically and foreshadowing .

Contributions of Descartes and Fermat

René Descartes introduced the foundational elements of the coordinate system in his 1637 treatise La Géométrie, an appendix to Discours de la méthode. He systematically employed two intersecting lines as fixed axes to locate points in the plane, allowing geometric problems to be translated into algebraic equations. Descartes denoted points using letters to represent their distances from the axes, such as assigning letters like a, b, or z to coordinates measured along perpendicular lines, which facilitated the algebraic description of curves. Additionally, he formulated the rule of signs, which bounds the number of positive real roots of a polynomial by the number of sign changes in its coefficients, aiding in the analysis of algebraic equations derived from geometric constructions. This rule, stated in La Géométrie, provided a practical tool for determining possible roots without solving equations fully, though it did not distinguish between real and complex roots. Independently, developed an early form of coordinate geometry around 1636 in his unpublished manuscript Introduction to Plane and Solid Loci. In this work, Fermat emphasized the representation of algebraic through equations relating distances from a fixed line (akin to an ) to a variable ordinate, enabling the study of loci as algebraic entities. His "ad lineam" method involved comparing values at adjacent points along a to determine tangents, maxima, and minima, relying on algebraic techniques to approximate properties without explicit coordinate pairs. Fermat's approach focused on generating and analyzing dynamically, such as parabolas and hyperbolas, by expressing ordinates as functions of abscissas in algebraic form, though he did not publish these ideas during his lifetime. The contributions of Descartes and Fermat differed in emphasis: Descartes sought to reduce all geometric problems to algebraic manipulation using a rigid system of intersecting axes, prioritizing a universal framework for construction and proof, whereas Fermat's methods centered on for optimization problems like maxima and minima, treating curves as generated by algebraic relations along a single reference line. This contrast highlighted Descartes' algebraic universality against Fermat's applied analytic techniques for curve properties. A priority dispute arose between them over the invention of , fueled by Fermat's earlier ideas circulated privately. , a key intermediary in the Parisian mathematical circle, disseminated Fermat's 1636 letter on tangents and mediated the exchange, but Descartes' insistence on his primacy prevented resolution, despite Mersenne's efforts to foster collaboration. Fermat's work reached wider audiences posthumously in 1679, while Descartes' published established the standard framework.

Evolution and naming conventions

The , initially developed in the , underwent significant refinement in the through the work of Leonhard Euler, who employed and helped popularize the notation of ordered pairs ([x, y](/page/X&Y)) for representing points in the plane through his analytical works, including his 1768 treatise Institutionum calculi integralis. This innovation facilitated clearer algebraic manipulation of geometric problems and was instrumental in advancing analytical methods. Euler's approach built on earlier foundations by emphasizing the systematic use of variables for coordinates, enabling more precise calculations in differential equations and curve analysis. The system also saw adoption in the burgeoning field of during the late 17th and early 18th centuries, where and employed to synthesize geometric and algebraic techniques, laying the groundwork for their independent developments of infinitesimal methods. Newton's (1687) utilized coordinate-like representations to describe trajectories and forces, while Leibniz's analytical framework integrated coordinates to handle tangents and areas under curves, transforming into a branch of . By the 19th century, the coordinate system was formalized and extended to three dimensions and beyond by mathematicians such as , whose work in and incorporated multidimensional coordinates to model surfaces and spatial measurements. This period also marked the widespread adoption of the term "Cartesian coordinates" in mathematical literature, primarily to honor for his foundational role in , even as Pierre de Fermat's parallel contributions were acknowledged but less emphasized in naming conventions. Key textbooks, including Adrien-Marie Legendre's Éléments de géométrie (1794), integrated geometric principles that complemented coordinate methods, promoting their pedagogical standardization across . Further evolution occurred through the late 19th-century development of vector analysis by J. Willard Gibbs and , who relied on Cartesian coordinates as the foundational framework for defining vector operations like and products, enabling concise formulations in and . Their independent systems, detailed in Gibbs's lecture notes (circa ) and Heaviside's Electromagnetic Theory (1893), emphasized orthogonal axes for practical computations. In the 20th century, international bodies like the (ISO) established conventions for axes and notation, as outlined in ISO 80000-2 (first published in 2009, building on earlier 20th-century standards), ensuring uniform representation of coordinates in scientific and engineering contexts.

Fundamental Description

One-dimensional coordinates

The one-dimensional Cartesian coordinate system is the simplest form of the Cartesian framework, consisting of a single straight line serving as the axis, with points on this line uniquely identified by a single known as the coordinate. This line, often called the , has a designated point at 0, from which all other positions are measured. The coordinate of a point indicates its signed from the origin along the line, establishing a direct correspondence between real numbers and locations on the axis. Key to this system are the conventions for positive and negative directions, which introduce directionality and ordering. Points with positive coordinates lie to the right of the , representing distances in the positive direction, while points with negative coordinates lie to the left, indicating distances in the negative direction. This setup imposes a on the points: for any two distinct points with coordinates a and b, if a < b, then the point with coordinate a is positioned to the left of the point with coordinate b. The number line representation visually embodies these properties, allowing for intuitive comprehension of inequalities and sequences of real numbers. For instance, the point denoted as x = 3 is located 3 units to the right of the origin, while x = -2 is 2 units to the left. Such examples illustrate how the coordinate directly quantifies position without requiring additional geometric constructs. This one-dimensional system establishes the core principle of numerical labeling of positions, providing essential prerequisite knowledge for extending the to multiple axes in higher-dimensional spaces.

Two-dimensional coordinates

The two-dimensional Cartesian coordinate system builds upon the one-dimensional number line by introducing a second axis perpendicular to the first, creating a plane for locating points using ordered pairs. It consists of two orthogonal axes: the horizontal x-axis and the vertical y-axis, which intersect at a common point called the origin, denoted as (0,0). Each point in the plane is uniquely specified by an ordered pair (x, y), where x represents the signed distance from the origin along the x-axis (positive to the right, negative to the left) and y represents the signed distance along the y-axis (positive upward, negative downward). This convention follows the standard orientation where the positive x-direction points right and the positive y-direction points up, ensuring a consistent right-handed system for planar descriptions. Geometrically, the Cartesian plane corresponds to the set \mathbb{R}^2, the collection of all ordered pairs of real numbers, with the axes serving as reference lines and their parallels forming a rectangular grid that divides the plane into unit squares for precise positioning. For instance, the point (2,3) is found by moving 2 units right along the x-axis and then 3 units up parallel to the y-axis from the origin. Similarly, graphing a function such as y = x involves plotting points like (0,0), (1,1), and (-1,-1), then connecting them to form a straight line through the plane.

Three-dimensional coordinates

The three-dimensional Cartesian coordinate system extends the two-dimensional plane by adding a third mutually perpendicular axis, the z-axis, which is orthogonal to the xy-plane. Points in three-dimensional Euclidean space, denoted \mathbb{R}^3, are specified by ordered triples (x, y, z), where x measures distance along the x-axis from the yz-plane, y along the y-axis from the xz-plane, and z along the z-axis from the xy-plane. The three axes intersect at the origin (0, 0, 0) and are oriented using the right-hand rule for consistency: point the thumb of the right hand in the positive x-direction, the index finger in the positive y-direction, and the middle finger will align with the positive z-direction. This convention ensures a right-handed system, facilitating spatial visualization where the xy-plane forms a horizontal base and the z-axis extends vertically. Key to understanding three-dimensional space are the planes parallel to the coordinate planes: those defined by x = k (parallel to the yz-plane), y = k (parallel to the xz-plane), and z = k (parallel to the xy-plane), where k is a constant. These planes, along with the coordinate planes themselves, divide \mathbb{R}^3 into eight regions known as octants, labeled I through VIII based on the signs of the coordinates (e.g., octant I for x > 0, y > 0, z > 0). For example, the point (1, 2, 3) lies in the first octant: from the , move 1 unit along the positive x-axis, then 2 units parallel to the positive y-axis, and finally 3 units parallel to the positive z-axis. Such coordinates are fundamental for locating vertices in simple solids, like the corners of a rectangular prism in a grid, enabling descriptions of volumes and positions in physical and geometric models.

Extensions to Higher Dimensions

n-dimensional coordinate systems

In the n-dimensional Cartesian coordinate system, points in the \mathbb{R}^n are represented as ordered n-tuples (x_1, x_2, \dots, x_n) of real numbers, where each x_i specifies the signed distance from the along the corresponding axis. This framework extends the lower-dimensional cases by assuming n mutually orthogonal directions, allowing unique identification of any point through its coordinates relative to a fixed . The coordinate system is defined with respect to an orthonormal basis consisting of the standard basis vectors \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n, where \mathbf{e}_i is the vector with 1 in the i-th position and 0 elsewhere; any point \mathbf{x} can thus be expressed as \mathbf{x} = \sum_{i=1}^n x_i \mathbf{e}_i. A key property is the induced Euclidean inner product, defined as \langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^n u_i v_i for vectors \mathbf{u} = (u_1, \dots, u_n) and \mathbf{v} = (v_1, \dots, v_n), which measures the cosine of the angle between them and enables the definition of distances and orthogonality. This inner product ensures that \mathbb{R}^n forms a complete inner product space, supporting notions of length via the norm \|\mathbf{x}\| = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}. Applications of n-dimensional coordinates abound in linear algebra, where linear transformations between subspaces of \mathbb{R}^n and \mathbb{R}^m are represented by m \times n matrices that operate on coordinate vectors to produce new coordinates. In data science, high-dimensional feature vectors—such as those encoding multiple attributes of data points—facilitate machine learning tasks like classification and clustering by embedding observations into \mathbb{R}^n for algorithmic processing. A example is (1, 2, 3, 4) in \mathbb{R}^4, which lies at distances 1, 2, 3, and 4 along its respective axes from the . Visualizing n-dimensional structures, such as the 4D () bounded by points where each coordinate is 0 or 1, is inherently challenging, as it exceeds three-dimensional perception and typically requires techniques like projections onto lower spaces to convey its 8 cubic cells and 16 vertices.

Generalizations beyond Euclidean spaces

Curvilinear coordinate systems extend the Cartesian framework by allowing coordinates to vary nonlinearly, adapting to the symmetry of the problem while locally approximating . In such systems, the coordinate surfaces intersect, and the describes the geometry of infinitesimal displacements. For orthogonal , the is given by ds^2 = h_1^2 du_1^2 + h_2^2 du_2^2 + h_3^2 du_3^2, where h_i are scale factors relating the curvilinear differentials to Cartesian ones via d\mathbf{r} = h_1 du_1 \hat{\mathbf{u}}_1 + h_2 du_2 \hat{\mathbf{u}}_2 + h_3 du_3 \hat{\mathbf{u}}_3. Common examples include polar coordinates in two dimensions, where (r, \theta) relate to Cartesian (x, y) by x = r \cos \theta, y = r \sin \theta, with scale factors h_r = 1 and h_\theta = r, providing a local tangent basis that aligns with Cartesian axes away from the origin. Cylindrical coordinates (s, \phi, z) extend this to three dimensions by adding a linear z-axis, suitable for axisymmetric problems like fluid flow around cylinders, where the metric is ds^2 = ds^2 + s^2 d\phi^2 + dz^2. Spherical coordinates (r, \theta, \phi) further generalize for spherical symmetry, with x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi, z = r \cos \theta, and scale factors h_r = 1, h_\theta = r, h_\phi = r \sin \theta, approximating Cartesian coordinates in small regions where curvature is negligible. These systems use the to transform tensors and vectors between curvilinear and Cartesian bases, ensuring physical laws remain covariant. In non- spaces, Cartesian-like coordinates are adapted to curved geometries, where the deviates from the flat form, and connections such as account for the in defining —the analogs of straight lines. In , the uses coordinates (x, y) within a unit disk with ds^2 = \frac{4(dx^2 + dy^2)}{(1 - x^2 - y^2)^2}, where are circular arcs orthogonal to the boundary; , derived from the , are \Gamma^x_{xx} = \frac{2x}{1 - r^2}, \Gamma^x_{xy} = \frac{2y}{1 - r^2}, and similarly for others, enabling computation of equations \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0. , often realized on a , employs spherical coordinates with the round ds^2 = r^2 (d\theta^2 + \sin^2 \theta d\phi^2), where are great circles; the non-vanishing include \Gamma^\theta_{\phi\phi} = -\sin \theta \cos \theta and \Gamma^\phi_{\theta\phi} = \cot \theta, reflecting the positive . These symbols facilitate the parallel transport and flow in spaces of constant negative () or positive () . In modern applications, such as , the posits that locally, in a sufficiently small , can be described by Cartesian coordinates in an inertial frame, where the approximates the Minkowski form \eta_{\mu\nu} = \operatorname{diag}(-1,1,1,1) and vanish, mimicking . This local Cartesian approximation holds in freely falling frames, allowing tidal effects to be the only deviation from flatness. In manifold theory, smooth manifolds are covered by coordinate charts that locally resemble \mathbb{R}^n, with transition maps ensuring compatibility; these charts provide local Cartesian-like coordinates for defining spaces and structures. Tensor coordinates in physics, particularly in and field theory, generalize Cartesian systems via the to handle arbitrary geometries, enabling the expression of physical laws in covariant form. However, curvilinear systems like spherical coordinates exhibit singularities, such as at the poles where \theta = 0 or \pi, causing the azimuthal coordinate \phi to become ill-defined and scale factor h_\phi = r \sin \theta \to 0, leading to coordinate breakdowns that require careful handling in computations, often via regularization or alternative charts.

Notations and Conventions

Standard notation for points and axes

In the standard Cartesian coordinate system, points in two-dimensional space are denoted using ordered pairs of real numbers enclosed in parentheses, such as (x, y), where x represents the horizontal displacement from the origin along the x-axis and y the vertical displacement along the y-axis./10%3A_Vectors/10.01%3A_Introduction_to_Cartesian_Coordinates_in_Space) This notation extends to three dimensions as ordered triples (x, y, z), with z indicating displacement along the perpendicular z-axis. For higher dimensions, points are expressed as ordered n-tuples (x_1, x_2, \dots, x_n), where the subscripts distinguish coordinates along successive orthogonal axes. The axes themselves are conventionally labeled as x, y, and z (or more generally x_1, x_2, x_3, \dots) in three or more dimensions, originating from a fixed point called the origin, denoted O with coordinates (0, 0) in 2D or (0, 0, 0) in 3D. Position vectors from the origin to a point are often represented in boldface or with arrows, such as \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}, where \mathbf{i}, \mathbf{j}, and \mathbf{k} are the unit vectors along the respective axes, each having magnitude 1 and pointing in the positive direction. These unit vectors form an for the space, enabling the decomposition of any into components aligned with the axes. Variations in notation exist across contexts; while parentheses are standard in mathematics for coordinate tuples to distinguish them from intervals or other structures, square brackets [x, y] are sometimes used in computer science to represent arrays or coordinate lists, reflecting programming conventions for data structures./06%3A_Appendices/6.01%3A_A_Appendices/6.1.07%3A_A.7_ISO_Coordinate_System_Notation) The International Organization for Standardization (ISO 80000-2) endorses the parenthetical form for mathematical coordinates and specifies lowercase letters like x, y, z for axes in Euclidean spaces, promoting consistency in . Historically, René Descartes initially employed letters from the end of the alphabet (such as x, y, z) for unknowns in geometric equations in his 1637 work La Géométrie, marking an early use of such symbols for coordinates in analytic geometry. Leonhard Euler later popularized the systematic use of x, y, z as variables for coordinates and functions in the 18th century, shifting from Descartes' more ad hoc lettering to a standardized convention that facilitated broader algebraic treatments of geometry.

Quadrants, octants, and spatial divisions

In the two-dimensional Cartesian plane, the intersecting x- and y-axes divide the infinite plane into four regions known as , each defined by the specific combination of positive or negative signs for the x- and y-coordinates of points within them. Quadrant I encompasses points where both coordinates are positive (x > 0, y > 0), Quadrant II includes points with negative x and positive y (x < 0, y > 0), Quadrant III covers points where both are negative (x < 0, y < 0), and Quadrant IV contains points with positive x and negative y (x > 0, y < 0). The axes serve as boundaries separating these , with points on the axes themselves not belonging to any quadrant. Quadrants are conventionally numbered using Roman numerals I through IV, beginning with Quadrant I in the upper-right region and proceeding counterclockwise around the origin. Extending to three dimensions, the three mutually perpendicular coordinate planes (xy, yz, and xz) partition Euclidean space into eight regions called octants, analogous to quadrants but incorporating the sign of the -coordinate alongside x and y. Each octant is identified by a triplet of signs, such as the first octant where all coordinates are positive (x > 0, y > 0, z > 0), the second where x is negative and y, z positive (x < 0, y > 0, z > 0), and so on through all eight combinations like (-, -, +) or (+, +, -). The coordinate planes act as boundaries, excluding points on them from the open octants. Unlike quadrants, octants lack a universally standardized numbering , though the all-positive region is commonly designated as the first octant, with subsequent numbering often using in a systematic progression based on sign changes, typically starting from the positive directions and cycling through variations in x, y, and . In higher-dimensional Euclidean spaces, this partitioning generalizes to n dimensions, where the n coordinate hyperplanes divide \mathbb{R}^n into $2^n regions termed , each corresponding to a unique of sign combinations for the n coordinates (e.g., (+, -, +, \dots)). extend the of quadrants (for n=2) and octants (for n=3), providing a way to classify points based on the of each coordinate relative to the . In fields like optimization, play a key role in defining feasible regions; for instance, in , the nonnegative —where all variables satisfy x_i \geq 0 for i=1 to n—often bounds the feasible set of solutions satisfying nonnegativity constraints. These spatial divisions facilitate and by assigning descriptive labels to subsets of space, aiding in geometric interpretations and problem-solving across and applications.

Geometric Formulas in the Plane

Distance between points

In the two-dimensional Cartesian plane, the straight-line distance between two points P = (x_1, y_1) and Q = (x_2, y_2) is calculated by considering the formed by the horizontal displacement |x_2 - x_1| and the vertical displacement |y_2 - y_1|. By the , this distance d(P, Q) is the length, given by the formula d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. This derivation directly extends the geometric principle of the to coordinate geometry, where the axes provide the legs of the . For example, the distance between the points (1, 2) and (4, 6) is d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. This illustrates how the formula quantifies spatial separation in the plane. The Euclidean distance generalizes naturally to n-dimensional Euclidean space \mathbb{R}^n, where points are represented as vectors \mathbf{x} = (x_1, \dots, x_n) and \mathbf{y} = (y_1, \dots, y_n). The distance is defined as the Euclidean norm of the difference vector \mathbf{x} - \mathbf{y}: d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}. This formula arises from the inner product structure of \mathbb{R}^n, where the norm \|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} extends the Pythagorean theorem to orthogonal components in higher dimensions. The function serves as a on \mathbb{R}^n, satisfying key properties that ensure it measures "distance" consistently. It is non-negative, with d(\mathbf{x}, \mathbf{y}) \geq 0 and equality \mathbf{x} = \mathbf{y} (). It is symmetric, so d(\mathbf{x}, \mathbf{y}) = d(\mathbf{y}, \mathbf{x}). Finally, it obeys the : d(\mathbf{x}, \mathbf{z}) \leq d(\mathbf{x}, \mathbf{y}) + d(\mathbf{y}, \mathbf{z}) for any points \mathbf{x}, \mathbf{y}, \mathbf{z}. These axioms confirm its role as the standard inducing the usual on .

Midpoint and line equations

In the Cartesian plane, the midpoint of a connecting two points (x_1, y_1) and (x_2, y_2) is the point that divides the segment into two equal parts, obtained by averaging the coordinates of the endpoints./11:_Conics/11.02:_Distance_and_Midpoint_Formulas_and_Circles) The coordinates of this midpoint M are given by the formula: M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) This derivation follows from the geometric property that the midpoint lies equidistant from both endpoints, which can be verified using the distance formula; the average ensures equal distances along each axis. For example, the midpoint of the segment joining (0, 0) and (2, 4) is (1, 2). This concept generalizes to higher dimensions, where the midpoint of points in n-dimensional space is the arithmetic mean of their coordinates, forming the centroid for equal weighting./11:_Conics/11.02:_Distance_and_Midpoint_Formulas_and_Circles) Lines in the Cartesian plane can be represented algebraically in several forms, each suited to different contexts such as graphing or parametric traversal. The slope-intercept form expresses the line as y = mx + b, where m is the (rise over run) and b is the y-intercept. The point-slope form, useful when a point and are known, is (y - y_1) = m(x - x_1), derived by rearranging the definition between two points./03:_Linear_Functions/3.04:_The_Point-Slope_Form_of_a_Line) Parametric equations provide a -based , parameterizing the line through a point (x_1, y_1) in the of a (a, b) as: x = x_1 + at, \quad y = y_1 + bt where t is a scalar varying over the reals. This form arises from adding scalar multiples of the to the of the point, capturing the line's infinite extent. For the example points (0, 0) and (2, 4), the is (2, 4), yielding parametric equations x = 0 + 2t, y = 0 + 4t, or equivalently in slope-intercept form y = 2x.

Transformations of Coordinates

Rigid Euclidean transformations

Rigid Euclidean transformations, also known as isometries of the plane, are distance-preserving mappings that maintain the geometric structure of figures in the Cartesian coordinate system. These transformations include translations, rotations, and reflections, each of which can be expressed using coordinate formulas or matrix representations. They form the E(2), the group of all rigid motions of the plane under composition. A shifts every point in the by a fixed (h, k), given by the formulas x' = x + h and y' = y + k. In affine matrix form, this is represented as the augmented by the translation vector: \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h \\ k \end{pmatrix}. Translations preserve and do not alter distances or angles between points. Rotation by an angle \theta counterclockwise around the transforms coordinates via x' = x \cos \theta - y \sin \theta and y' = x \sin \theta + y \cos \theta. The corresponding is \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}. For example, rotating the point (1, 0) by 90° (\theta = \pi/2) yields x' = 1 \cdot 0 - 0 \cdot 1 = 0 and y' = 1 \cdot 1 + 0 \cdot 0 = 1, resulting in (0, 1). Rotations also preserve distances and , maintaining the of the figure. Reflections are orientation-reversing isometries. Reflection over the x-axis maps (x, y) to (x, -y), with matrix \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. Reflection over the line y = x swaps coordinates: (x, y) to (y, x), represented by \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. Like other rigid transformations, reflections preserve distances and angles but reverse orientation. The composition of these isometries generates the full Euclidean group E(2), which includes both orientation-preserving (translations and rotations) and orientation-reversing (reflections) elements.

Affine transformations including scaling and shearing

Affine transformations in the Cartesian coordinate system are compositions of linear transformations and translations that preserve and ratios of distances along lines, straight lines to straight lines while maintaining parallelism. In two dimensions, an affine transformation can be expressed as \begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} b_x \\ b_y \end{pmatrix}, where A is a $2 \times 2 representing the linear part and \mathbf{b} = (b_x, b_y) is the vector. This form encompasses a broad class of s, including those that distort shapes without preserving distances or angles, in contrast to rigid transformations that maintain both. Scaling is a specific type of that multiplies coordinates by constants, either uniformly or non-uniformly, to enlarge or reduce figures relative to the . Uniform by a factor k transforms a point (x, y) to (k x, k y) and is represented by the A = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}. Non-uniform applies different factors to each , such as (k_x x, k_y y), using the matrix A = \begin{pmatrix} k_x & 0 \\ 0 & k_y \end{pmatrix}, which stretches or compresses along the x- and y-directions independently. Shearing transformations skew figures by adding a multiple of one coordinate to the other, preserving area but altering angles. A horizontal shear by factor m maps (x, y) to (x + m y, y) and uses the matrix A = \begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix}, sliding points parallel to the x-axis based on their y-coordinate. Conversely, a vertical shear by m transforms (x, y) to (x, y + m x) with A = \begin{pmatrix} 1 & 0 \\ m & 1 \end{pmatrix}, shifting along the y-axis according to the x-coordinate. For example, applying a horizontal shear with m=1 to the point (1,1) yields (2,1), demonstrating the skewing effect. Key properties of affine transformations include the preservation of straight lines and the ratios of distances between points on those lines, ensuring that combinations of points remain after mapping. The of the linear A, denoted \det(A), scales areas by its absolute value: if \det(A) = k, then the area of a is multiplied by |k| under the . These characteristics make affine transformations fundamental for modeling deformations in and while tying back to the Cartesian framework's emphasis on linear structure.

Orientation and Handedness

Orientation in two dimensions

In two dimensions, the Cartesian coordinate system establishes a positive through the standard basis vectors \mathbf{e}_1 = (1, 0) and \mathbf{e}_2 = (0, 1), where the spans the plane in a counterclockwise manner. This convention aligns with the adapted to the plane, treating counterclockwise traversal from the positive x-axis to the positive y-axis as the positive direction. Linear transformations in the plane, represented by a $2 \times 2 matrix A, either preserve or reverse this orientation based on the sign of their determinant \det(A). A transformation preserves positive orientation if \det(A) > 0, as seen in rotations and uniform scalings, which maintain the counterclockwise order of basis vectors; conversely, \det(A) < 0 reverses orientation, as in reflections, which swap the order to clockwise. Translations, being affine rather than linear, inherently preserve orientation by not altering relative directions. For example, a counterclockwise rotation by 90° has the matrix \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, with \det(A) = 1 > 0, preserving by mapping \mathbf{e}_1 to \mathbf{e}_2 and \mathbf{e}_2 to -\mathbf{e}_1 while keeping the counterclockwise span. In contrast, over the x-axis uses the matrix \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, with \det(A) = -1 < 0, reversing orientation by flipping \mathbf{e}_2 to -\mathbf{e}_2. This determinant test extends to applications in the complex plane, where the Cartesian coordinates identify \mathbb{R}^2 with \mathbb{C}. Multiplication by a nonzero complex number re^{i\theta} corresponds to scaling by r > 0 and counterclockwise rotation by \theta, both preserving orientation since the associated matrix \begin{pmatrix} r \cos \theta & -r \sin \theta \\ r \sin \theta & r \cos \theta \end{pmatrix} has determinant r^2 > 0.

Handedness in three dimensions

In three dimensions, the handedness of a Cartesian coordinate system refers to the that distinguishes right-handed from left-handed configurations, extending the concept of from two dimensions to volumetric . A right-handed system aligns with the conventional positive , while a left-handed one reverses it, affecting the interpretation of directions in operations and transformations./04%3A_R/4.09%3A_The_Cross_Product) The defines the positive orientation in a right-handed Cartesian coordinate system. To apply it, point the thumb of the right hand in the direction of the positive x-axis, the along the positive y-axis, and the will then point along the positive z-axis. This rule ensures consistency in assigning directions to axes and is fundamental for defining the vectors \mathbf{e}_1 = (1,0,0), \mathbf{e}_2 = (0,1,0), and \mathbf{e}_3 = (0,0,1) as right-handed. Handedness can be quantitatively determined using the scalar of three vectors \mathbf{u}, \mathbf{v}, and \mathbf{w}, defined as [\mathbf{u}, \mathbf{v}, \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}). This product equals the of the matrix whose columns are \mathbf{u}, \mathbf{v}, and \mathbf{w}; if the value is positive, the vectors form a , and if negative, a left-handed one. The represents the signed volume of the spanned by the vectors, with the sign indicating the relative to the ./04%3A_R/4.09%3A_The_Cross_Product) Improper transformations, such as spatial inversion (replacing \mathbf{r} = (x,y,z) with -\mathbf{r} = (-x,-y,-z)) or mirror reflections across a , reverse the handedness of the coordinate system. These operations are represented by matrices with -1, which flip the sign of the scalar triple product and thus convert a right-handed system to left-handed or vice versa. In contrast, proper rotations ( +1) preserve . In physics, handedness plays a key role in applications like the right-hand screw rule for angular momentum, where curling the fingers of the right hand in the direction of rotation points the thumb along the axis of the angular momentum vector \mathbf{L} = \mathbf{r} \times \mathbf{p}. This convention ensures consistent directionality in rotational dynamics. In quantum mechanics, parity refers to the symmetry under spatial inversion, which reverses handedness; the discovery of parity violation in weak interactions, such as beta decay, demonstrated that fundamental physical laws are not invariant under this transformation, impacting particle physics models./9%3A_Rotational_Kinematics_Angular_Momentum_and_Energy/9.7%3A_Vector_Nature_of_Rotational_Kinematics).pdf) For example, the standard basis \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 forms a right-handed system, as its scalar triple product is \det \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = 1 > 0. Swapping the y- and z-axes yields \mathbf{e}_1 = (1,0,0), \mathbf{e}_2 = (0,0,1), \mathbf{e}_3 = (0,1,0), with determinant -1 < 0, making it left-handed./04%3A_R/4.09%3A_The_Cross_Product)

Vector Representation

Vectors in the standard Cartesian basis

In the standard Cartesian coordinate system, a vector \mathbf{v} is represented as a linear combination of the orthonormal basis vectors \hat{\mathbf{i}}, \hat{\mathbf{j}}, and \hat{\mathbf{k}}, which point along the positive x-, y-, and z-axes, respectively. These basis vectors are unit vectors with magnitude 1 and are mutually perpendicular, satisfying \hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = 0, \hat{\mathbf{i}} \cdot \hat{\mathbf{k}} = 0, and \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = 0. Thus, \mathbf{v} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}, where x, y, and z are the scalar components (coordinates) of \mathbf{v} in this basis, often denoted compactly as the ordered tuple (x, y, z). The orthonormality of the basis simplifies vector operations, as the coordinates directly correspond to projections onto the axes. Vector addition is performed component-wise: if \mathbf{w} = (u, v, w), then \mathbf{v} + \mathbf{w} = (x + u, y + v, z + w). The dot product, an inner product yielding a scalar, is \mathbf{v} \cdot \mathbf{w} = x u + y v + z w = \sum_{i=1}^3 x_i w_i, where x_1 = x, x_2 = y, x_3 = z, and similarly for \mathbf{w}. The magnitude (Euclidean norm) of \mathbf{v} is \|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{x^2 + y^2 + z^2}. To illustrate, consider \mathbf{v} = (1, 2, 3) and \mathbf{w} = (4, 5, 6). Then \mathbf{v} + \mathbf{w} = (5, 7, 9) and \mathbf{v} \cdot \mathbf{w} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32. The is \|\mathbf{v}\| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}. These operations rely on the fixed ; changing to a different basis requires transforming the coordinates via to preserve the vector's geometric properties.

Coordinate transformations for vectors

In the context of Cartesian coordinate systems, transformations under changes of coordinates can be distinguished as active or passive. An active physically rotates or moves the in space while keeping the coordinate basis fixed, resulting in new components for the same . In contrast, a passive relabels the coordinates by rotating the basis itself, yielding new components for the without altering its physical position. When the basis is rotated, the components of a vector \mathbf{v} in the new coordinate system are given by \mathbf{v}' = R \mathbf{v}, where R is an orthogonal rotation matrix satisfying R^T R = I and \det R = \pm 1. The condition R^T R = I ensures that the transformation preserves lengths and angles, maintaining the Euclidean structure of the space, while \det R = 1 corresponds to proper rotations (preserving orientation) and \det R = -1 to improper ones (including reflections). For passive transformations, the inverse relation \mathbf{v} = R \mathbf{v}' applies, reflecting the change in basis description. In three dimensions, rotations are commonly parameterized using , which compose three successive rotations about specific axes (e.g., z-x-z convention), or quaternions, which represent rotations as unit quaternions avoiding singularities like inherent in Euler angles. Under such rotations, the \mathbf{u} \times \mathbf{v} transforms to R (\mathbf{u} \times \mathbf{v}) for proper rotations (\det R = 1), preserving the and ; however, improper rotations (\det R = -1) reverse the cross product's direction, flipping the handedness of the system. These transformations find applications in , where viewing transformations convert world coordinates to camera coordinates using matrices to simulate observer orientation. In , Lorentz transformations extend this framework to four-vectors in , mixing time and space components while preserving the Minkowski metric, though they depart from pure Cartesian rotations. For example, consider a \mathbf{v} = (1, 0, 0) in a right-handed Cartesian basis. A 90° around the z-axis yields the matrix R = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, so \mathbf{v}' = R \mathbf{v} = (0, 1, 0), illustrating the passive relabeling of components.