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While locating the polar coordinates of a point, why do we take the initial line as a directed line? Doesn't it suffice to mention that the angle that it makes will be measured with the positive direction of the $x$ axis?
Couldn't we just take a line segment?

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  • $\begingroup$ What do you think "a directed line" (?) means? A directed vector is a segment of line with a definite direction, however you define that. $\endgroup$
    – DonAntonio
    Apr 15, 2017 at 9:51
  • $\begingroup$ I never make any mention to directed lines when I look at polar coordinates, I do just say the argument is the angle it makes with the positive $x$-axis. $\endgroup$
    – Bilbottom
    Apr 15, 2017 at 10:31
  • $\begingroup$ @DonAntonio I know what a vector is. It's a directed line segment. But could we do without using a vector while specifying a polar coordinate? Could we not talk of a directed line segment and simply use a line segment? $\endgroup$ Apr 15, 2017 at 11:02
  • $\begingroup$ @BillWallis Most textbooks start by using the angle made by a directed line segment to define the argument. What I'm asking is couldn't we do without assigning a direction to the initial line segment? We are already saying that the angle is made with the positive direction of the $x$ axis. $\endgroup$ Apr 15, 2017 at 11:04

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To start off, I'd like to ask this : How do we define a vector completely?

We usually do so by defining three things.

  1. It's sense
  2. The angle it makes with a certain reference axis.
  3. It's point of application

What I can infer is, you intend to ask that if we have defined the sense of the vector, why is there a need to specify the angle conventions, and vice versa.

I believe that the word directed segment itself implies that one of these things have to be defined. What the text means to say is that we take a segment, we measure the angle from the positive x, which makes the segment into a directed one.

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