Cartesian and polar coordinates in three dimensions | Mathematics for Grade 12 PDF Download

Introduction

• When each point on a plane of a two-dimensional coordinate system is decided by a distance from a reference point and an angle is taken from a reference direction, it is known as the polar coordinate system.
• Pole = The reference point
  Polar axis = the line segment ray from the pole in the reference direction
• In the polar coordinate system, the origin is called a pole.
  
• Here, instead of representing the point as (x, y), we can express it as a polar coordinate (r, θ).
  Where the value of r can be negative. The value of angle changes based on the quadrant in which the r lies.

Note:

In the Cartesian coordinate system, the distance of a point from the y-axis is called its x-coordinate and the distance of a point from the x-axis is called its y-coordinate.

Polar grid

Polar grid with different angles as shown below:

Also, π radians are equal to 360°.

Polar Coordinates Formula

We can write an infinite number of polar coordinates for one coordinate point, using the formula

• (r, θ+2πn) or (-r, θ+(2n+1)π), where n is an integer.
• The value of θ is positive if measured counterclockwise.
• The value of θ is negative if measured clockwise.
• The value of r is positive if laid off at the terminal side of θ.
• The value of r is negative if laid off at the prolongation through the origin from the terminal side of θ.

Note:

The side where the angle starts is called the initial side and the ray where the measurement of the angle stops is called the terminal side.

Cartesian to Polar Coordinates

x = r cos θ
y = r sin θ

Finding r and θ using x and y:

3D Polar Coordinates

• 3d polar coordinates or spherical coordinates will have three parameters: distance from the origin and two angles.
• The 3d-polar coordinate can be written as (r, Φ, θ).
• Here,
  R = distance of from the origin
  Φ = the reference angle from XY-plane (in a counter-clockwise direction from the x-axis)
  θ = the reference angle from z-axis
  

Polar Coordinates Examples

Example 1: Convert the polar coordinate (4, π/2) to a rectangular point.

Sol:
Given,

We know that,

Hence, the rectangular coordinate of the point is (0, 4).

Example 2: Convert the rectangular or cartesian coordinates (2, 2) to polar coordinates.

Sol:
Given, (x, y)=(2, 2)

Hence, the polar coordinates are

Note:
We know that, the value of r can be negative too.
That means r = ±2√2
Also, tan⁻¹(1) will also take many values, such as

Hence, the possible polar coordinates of a point (2, 2) are