Cartesian coordinates

(noun)

The coordinates of a point measured from an origin along a horizontal axis from left to right (the xxx-axis) and along a vertical axis from bottom to top (the yyy-axis). 

Related Terms

  • ordered pair

Examples of Cartesian coordinates in the following topics:

  • The Cartesian System

    • The Cartesian coordinate system is used to visualize points on a graph by showing the points' distances from two axes.
    • A Cartesian coordinate system is used to graph points.
    • The Cartesian coordinate system is broken into four quadrants by the two axes.
    • The four quadrants of theCartesian coordinate system.
    • The Cartesian coordinate system with 4 points plotted, including the origin, at (0,0)(0,0)(0,0).
  • Converting Between Polar and Cartesian Coordinates

    • Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.
    • When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
    • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
    • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
    • Derive and use the formulae for converting between Polar and Cartesian coordinates
  • Polar Coordinates

    • This is called the Cartesian coordinate system.
    • The xyxyxy or Cartesian coordinate system is not always the easiest system to use for every problem.
    • Polar coordinates in rrr and θ\thetaθ can be converted to Cartesian coordinates xxx and yyy.
    • A set of polar coordinates.
    • The xxx Cartesian coordinate is given by rcosθr \cos \thetarcosθ and the yyy Cartesian coordinate is given by rsinθr \sin \thetarsinθ.
  • Vectors in Three Dimensions

    • The mathematical representation of a physical vector depends on the coordinate system used to describe it.
    • In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.
    • Typically in Cartesian coordinates, one considers primarily bound vectors.
    • A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin O=(0,0,0)O = (0,0,0)O=(0,0,0).
    • A vector in the 3D Cartesian space, showing the position of a point AAA represented by a black arrow.
  • Three-Dimensional Coordinate Systems

    • Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters xxx, yyy, and zzz.
    • Often, you will need to be able to convert from spherical to Cartesian, or the other way around.
    • This is a three dimensional space represented by a Cartesian coordinate system.
    • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.
    • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system
  • Cylindrical and Spherical Coordinates

    • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian xyxyxy-plane (with equation z=0z = 0z=0), and the cylindrical axis is the Cartesian zzz-axis.
    • Then the zzz coordinate is the same in both systems, and the correspondence between cylindrical (ρ,φ)(\rho,\varphi)(ρ,φ) and Cartesian (x,y)(x,y)(x,y) are the same as for polar coordinates, namely x=ρcosφ;y=ρsinφx = \rho \cos \varphi; \, y = \rho \sin \varphix=ρcosφ;y=ρsinφ.
    • The spherical coordinates (radius rrr, inclination θ\thetaθ, azimuth φ\varphiφ) of a point can be obtained from its Cartesian coordinates (xxx, yyy, zzz) by the formulae:
    • A cylindrical coordinate system with origin OOO, polar axis AAA, and longitudinal axis LLL.
  • Double Integrals in Polar Coordinates

    • In R2R^2R​2​​, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points P(x,y)P(x, y)P(x,y) in Cartesian coordinates switch to their respective points in polar coordinates.
    • The polar coordinates rrr and φ\varphiφ can be converted to the Cartesian coordinates xxx and yyy by using the trigonometric functions sine and cosine:
    • The Cartesian coordinates xxx and yyy can be converted to polar coordinates rrr and φ\varphiφ with r≥0r \geq 0r≥0 and φ\varphiφ in the interval $(−\pi, \pi]$:
    • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.
    • This figure illustrates graphically a transformation from cartesian to polar coordinates
  • Other Curves in Polar Coordinates

    • Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.
    • To graph in the rectangular coordinate system we construct a table of xxx and yyy  values.
    • To graph in the polar coordinate system we construct a table of rrr and θ\thetaθ values.
  • Unit Vectors and Multiplication by a Scalar

    • The unit vectors are different for different coordinates.
    • In Cartesian coordinates the directions are x and y usually denoted x^\hat{x}​x​^​​ and y^\hat{y}​y​^​​.
    • The unit vectors in Cartesian coordinates describe a circle known as the "unit circle" which has radius one.
    • This can be seen by taking all the possible vectors of length one at all the possible angles in this coordinate system and placing them on the coordinates.
  • Introduction to the Polar Coordinate System

    • The polar coordinate system is an alternate coordinate system where the two variables are rrr and θ\thetaθ, instead of xxx and yyy.
    • When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y)(x,y)(x,y) in the Cartesian coordinate plane.
    • The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis.
    • The radial coordinate is often denoted by rrr or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or ttt.
    • In green, the point with radial coordinate 333 and angular coordinate 606060 degrees or (3,60∘)(3,60^{\circ})(3,60​∘​​).
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