Polar Coordinates: Understanding Negative Distance r

[Peter G.]

Hi,

I am learning about Polar Coordinates and how they can be written in several equivalent ways.

I understand how you can add 360 to angles and use negative angles to represent the same point.

However, I have a very hard time understanding how you can write the same point but with a negative distance r.

So, the example in the book is r = 10 and θ = 30.

I really can't see how r = -10 and θ = 210 is equivalent.

Would anyone mind trying to give me some reasoning better than the one provided by my book?

Thanks!

 

[eumyang]

Hi,

I am learning about Polar Coordinates and how they can be written in several equivalent ways.

I understand how you can add 360° to angles and use negative angles to represent the same point.

However, I have a very hard time understanding how you can write the same point but with a negative distance r.

So, the example in the book is r = 10 and θ = 30°.

I really can't see how r = -10 and θ = 210° is equivalent.

Would anyone mind trying to give me some reasoning better than the one provided by my book?

Thanks!

(Sorry to nitpick, but you really need the degree symbols. Otherwise, I have to assume that you are in radians.)

Think of it this way. Let's use the point (8, 135°) as an example. Pretend that you are standing at the origin. θ = 135° means that you would turn and face towards the NW direction. r = 8 indicates that you would walk forward 8 units in the direction of 135°.

Now (-8, 315°) is an equivalent point. θ = 315° means that you would turn and face towards the SE direction. r = -8 indicates that you would walk backwards 8 units (r is negative), while still facing the SE direction. So you end in the same spot as (8, 135°).

 

[Peter G.]

Firstly, sorry for the degree symbol, I understand. Regarding the explanation, thanks! That is great!