Cartesian coordinates

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 $(0,0)$.

• 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 $xy$ or Cartesian coordinate system is not always the easiest system to use for every problem.
  • Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
  • A set of polar coordinates.
  • The $x$ Cartesian coordinate is given by $r \cos \theta$ and the $y$ Cartesian coordinate is given by $r \sin \theta$.

• 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 $O = (0,0,0)$.
  • A vector in the 3D Cartesian space, showing the position of a point $A$ 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 $x$, $y$, and $z$.
  • 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 $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
  • Then the $z$ coordinate is the same in both systems, and the correspondence between cylindrical $(\rho,\varphi)$ and Cartesian $(x,y)$ are the same as for polar coordinates, namely $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
  • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Double Integrals in Polar Coordinates

  • In $R^2$, 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 $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
  • The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:
  • The Cartesian coordinates $x$ and $y$ can be converted to polar coordinates $r$ and $\varphi$ with $r \geq 0$ and $\varphi$ 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 $x$ and $y$  values.
  • To graph in the polar coordinate system we construct a table of $r$ and $\theta$ 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 $\hat{x}$ and $\hat{y}$.
  • 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 $r$ and $\theta$, instead of $x$ and $y$.
  • When we think about plotting points in the plane, we usually think of rectangular coordinates $(x,y)$ 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 $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.
  • In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$.