Introduction To Spherical And Rectangular Coordinates
The sets of values that explain the location of a given point in space are called coordinates. In a three dimensional space, a point is uniquely defined by three coordinates. In this article, we learn about the relation between rectangular and spherical coordinate systems.
The cartesian system has an alternate name, that is, the rectangular system which can be obtained by drawing three lines in space. All three points cross a point that is commonly termed as the origin and are perpendicular to each other. The lines are called axes of the system. The three planes are XY, XZ and YZ planes that define each pair of axes.
The distance between any arbitrary point and the planes are the coordinates of that point.
A coordinate system with a fixed origin and a zenith direction is a spherical coordinate system. An imaginary point that is present directly above the origin is the zenith. Zenith’s direction is the direction from the origin to the zenith.
Every arbitrary point has three spherical coordinates namely the radius, the polar angle and azimuth angle. The radius is the distance between the origin and the point in a three-dimensional space. The polar angle is the angle from the zenith direction and the line which connects the point with the origin. The azimuth angle is the angle between the point’s orthogonal projection to the plane and a fixed reference direction on that plane.
What is the Spherical Coordinate System? In mathematics, a spherical coordinate system is a coordinate system for 3-dimensional space where the position of a point is specified by a spherical coordinate triplet (r,
In short, In spherical coordinate system, each component of the vector is the function of
What is Rectangular coordinate System? A coordinate system that specifies the location of every point uniquely with the specific set of coordinates. In two-dimensional plane, system is based on two perpendicular axes, x-axis and y-axis and in three-dimensional space, the system is based on three mutually perpendicular axes, x-axis, y-axis and z-axis.
Relation between the Rectangular Coordinate system and Spherical Coordinate system
Spherical to Rectangular
The relation between the rectangular coordinate system and the spherical coordinate system is
Conversely, these equations can be expressed so that r,
Rectangular to Spherical
r2 = x2 + y2 + z2,
tan Φ = y/x and
Dot Product in Spherical Coordinates
To find the desired component of a vector, we have to take the dot product of the vector and a unit vector in the desired direction.
If a vector A = A1x + A2y + A3z is in a rectangle coordinate system, then
In spherical coordinate system, unit vectors are
Example: Find an equation in spherical coordinates for the cone surface represented by rectangular equation, x2 + y2 = z2.
Solution: Substituting the values of x, y, and z, we have