Home Vectors in Math Vectors in Physics 2D Coordinates 3D Coordinates Representing Vector Quantities Adding Vectors Products of Vectors Vector Fields   Marianne Breinig The University of Tennessee, Department of Physics and Astronomy The Cartesian coordinate system is the most commonly used coordinate system.  In two dimensions, this system consists of a pair of lines on a flat surface or plane, that intersect at right angles.  The lines are called axes and the point at which they intersect is called the origin.  The axes are usually drawn horizontally and vertically and are referred to as the x- and y-axes, respectively. A point in the plane with coordinates (a, b) is a units to the right of the y axis and b units up from the x axis if a and b are positive numbers.  If a and b are both negative numbers, the point is a units to the left of the y axis and b units down from the x axis.  In the figure above point P1 has coordinates (3, 4), and point P2 has coordinates (-1, -3). In three-dimensional Cartesian coordinates, the z axis is added so that there are three axes all perpendicular to each other. In the polar coordinate system, each point in the plane is assigned coordinates (r, f) with respect to a fixed line in the plane called the axis and a point on that line called the pole.  For a point in the plane, the r-coordinate is the distance from the point to the pole, and the f-coordinate is the counterclockwise angle between the axis and a line joining the origin to the point,  The r-coordinate is always positive and the range of f is from 0 to 2p (360o).  To be able to transform from Cartesian to polar coordinates and vice versa, we let the axis of the polar coordinate system coincide with the x-axis of the Cartesian coordinate system and the pole coincide with the origin. In the figure above he Point P1 has polar coordinates (r1, f1) = (5, 53.1o), and the point P2 has polar coordinates (r2, f2) = (3.16, 251.6o ). The Transformation equations are: x = r cosf,     y = r sinf r = (x2 + y2)1/2,    f = tan-1(y/x) Cylindrical coordinates and spherical coordinates are two different extensions of polar coordinates to three dimensions. Note: You do NOT have to enter a User ID.