Distance between Two Points in Polar Co-ordinates


How to find the distance between two points in polar Co-ordinates?

Distance between Two Points in Polar Co-ordinates


Let OX be the initial line through the pole O of the polar system and (r₁, θ ₁) and (r₂, θ₂) the polar co-ordinates of the points P and Q respectively. Then, OP₁ = r₁, OQ = r₂, ∠XOP = θ₁ and ∠XOQ = θ₂, Therefore, ∠POQ = θ₂ – θ₁. 

From triangle POQ we get,

PQ² = OP² + OQ² – 2 ∙ OP ∙ OQ ∙ cos∠POQ

    = r₁² + r₂² – 2r₁ r₂ cos(θ₂ - θ₁)

Therefore, PQ = √[r₁² + r₂ ² - 2r₁ r₂ cos⁡(θ₂ - θ₁)].

Second Method: Let us choose origin and positive x-axis of the cartesian system as the pole and initial line respectively of the polar system. If (x₁, y₁) , (x₂, y₂) and (r₁, θ₁) (r₂, θ₂) be the respective Cartesian and polar co-ordinates of the points P and Q, then we shall have,

    x₁ = y₁ cos θ₁,     y₁ = r₁ sin θ₁

and


    x₂ = r₂ cos θ₂,     y₂ = r₂ sin θ₂.

Now, the distance between the points P and Q is

PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]

     = √[(r₂ cos θ₂ - r₁ cos θ₁)² + (r₂ sin θ₂ - r₂ sin θ₂)²]

     = √[r₂² cos² θ₂ + r₁ ² cos² θ₁ - 2 r₁r₂ cos θ₁ cos θ₂ + r₂² sin² θ₂ + r₁²sin² θ₁ - 2 r₁r₁ sin θ₁ sin θ₂]

     = √[r₂² + r₁² - 2r₁ r₂ Cos(θ₂ - θ₁)].



Example on distance between two points in polar Co-ordinates:

Find the length of the line-segment joining the points (4, 10°) and (2√3 ,40°).

Solution:

We know that the length of the line-segment joining the points (r₁, θ₁),and (r₂, θ₂), is

     √[ r₂² + r₁² - 2r₁ r₂ Cos(θ₂ - θ₁)].

Therefore, the length of the line-segment joining the given points

     = √{(4² + (2√3)² - 2 ∙ 4 ∙ 2√(3) Cos(40 ° - 10°)}

     = √(16 + 12 - 16√3 ∙ √3/2)

     = √(28 - 24)

     = √4

     = 2 units.

 Co-ordinate Geometry 




11 and 12 Grade Math 

From Distance between Two Points in Polar Co-ordinates to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Shifting of Digits in a Number |Exchanging the Digits to Another Place

    May 19, 24 06:35 PM

    Shifting of Digits in a Number
    What is the Effect of shifting of digits in a number? Let us observe two numbers 1528 and 5182. We see that the digits are the same, but places are different in these two numbers. Thus, if the digits…

    Read More

  2. Formation of Greatest and Smallest Numbers | Arranging the Numbers

    May 19, 24 03:36 PM

    Formation of Greatest and Smallest Numbers
    the greatest number is formed by arranging the given digits in descending order and the smallest number by arranging them in ascending order. The position of the digit at the extreme left of a number…

    Read More

  3. Formation of Numbers with the Given Digits |Making Numbers with Digits

    May 19, 24 03:19 PM

    In formation of numbers with the given digits we may say that a number is an arranged group of digits. Numbers may be formed with or without the repetition of digits.

    Read More

  4. Arranging Numbers | Ascending Order | Descending Order |Compare Digits

    May 19, 24 02:23 PM

    Arranging Numbers
    We know, while arranging numbers from the smallest number to the largest number, then the numbers are arranged in ascending order. Vice-versa while arranging numbers from the largest number to the sma…

    Read More

  5. Comparison of Numbers | Compare Numbers Rules | Examples of Comparison

    May 19, 24 01:26 PM

    Rules for Comparison of Numbers
    Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of digits, we start comparing the digits…

    Read More