Distance Between Two Points in Polar Co-ordinates

Coordinate systems are commonly used to detect the position of a point in a given plane. In this article, we take a different approach to locating a point in a plane using polar coordinate systems.

We will discuss two points in polar coordinates and use the distance formula to derive an equation to find the distance between them.

In any plane, there are two coordinate systems: The polar coordinate system and the Cartesian coordinate system.

The Cartesian coordinate systems are defined by unit vectors, whereas the polar coordinates are defined by radial unit vectors.

We will discuss the distance between two points in polar coordinates in this article.

Important Terms Related to Coordinates

Here are the definitions of the important terms used in the coming sections.

Unit Vector: The unit vector has a magnitude of one. It is given by â for a vector in the direction of a.

Radial Unit Vector: It tells the position of a point from the origin and a unit vector which is perpendicular to the direction along the radius (radial direction)

Polar Coordinates

Standard cartesian coordinates tell the horizontal and vertical distance between two points in a two-dimensional plane. It is given by the x-axis and y-axis. One of the coordinates is fixed in the plane concerning which the position of the other point is measured. So, a point in the cartesian system is given by (x,y).

The polar coordinates system uses two polar coordinates. One is the radial coordinate, and the other is the angle made by a radial coordinate in the required direction. Radial angle is measured in radians.

The radial vector is taken about the origin. It points away from the origin to show the direction of any point. The radial direction is given by a unit radial vector.

Different information is used to describe the position of an object in the plane.

Let r be the radial distance of a point A from the centre of the cartesian plane. Let ϕ be the direction or angle moving anti-clockwise from the positive axis to the negative axis.

Therefore, the polar coordinate of point A is given by (r, ϕ).

Polar coordinates are not unique like cartesian coordinates. The same points can be described in many ways using polar coordinates.

Relating Polar Coordinates with Cartesian Coordinates

Polar coordinates are also used to express the cartesian coordinates of the vector. The relation between the two for a point A is obtained by

x = r cos ϕ

y = r sin ϕ

r = √[x2 + y2]

tan ϕ = y/x

Polar coordinates describe an object that moves around a circle. It means that only one coordinate ϕ changes with time.

Now that we have discussed the basics of polar coordinates, we will now find the distance between two points in polar coordinates.

Two Points in Polar Co-ordinates

The radial distance is usually represented by the letter r, and the radial angle is given by ϕ or θ. We have used ϕ in this article. Alternatively, you can use θ. They both represent the radial angle.

So, the point in the polar coordinate is represented as (r, ϕ) or (r, θ).

Suppose if there are two points in a polar coordinate, say A and P, the point A is represented as (r1, ϕ1), and the other point is represented as (r2, ϕ2).

Distance Between Two Points in Polar Co-ordinates

We use the distance formula for cartesian coordinates of x and y.

To find the distance between two points in polar coordinates, we will use the distance formula.

The Law of Cosines is used to find the distance or length by the formula:

a2 = b2 + c2 – 2bc cos A

a = √( b2 + c2 – 2bc cos A)

To find the distance between, say, (r1, ϕ1) and (r2, ϕ2) we substitute

b = r1 and c = r2. A is the angle between the two radii and is given by (ϕ2 – ϕ1). So, the distance between the two points in polar coordinates is given by

d = √ [r12 + r22 – 2 × r1 × r2 × cos (ϕ2 – ϕ1)]

You can use the formula to solve the distance between two points in polar coordinates questions.

Two Points in Polar Coordinates Examples

Polar coordinates are used in mechanical systems. They help in calculations based on fields. Magnetic fields, electric fields, and temperature fields use polar coordinates to make calculations easier. By using these calculations, the efficiency of machines can be increased. It helps in better understanding electricity and magnetism to produce power. These are some of the two points in polar coordinates examples.

Conclusion

Polar coordinates are a useful way for easing calculations in power generation. The two points in the polar coordinate system have two coordinates, one corresponding to the cartesian plane and the other corresponding to the polar coordinate.

We have discussed the various aspects of polar coordinates in this article. We have given examples of real-life situations in which they are used. We have also related polar coordinates to cartesian coordinates. We also found the distance formula to find the distance between two points in polar coordinates.