Chapter 8, Problem 1RCC

To determine

To describe: the polar coordinates represents the position of a point in the plane .

Answer to Problem 1RCC

  $P\left(r,\theta \right)$ is defined to be the point that lies $\left|r\right|$ units from the pole in the direction opposite to that given by $\theta$ .

Explanation of Solution

Given:

Calculation:

Describe how polar coordinates represent the position of a point in the plane.

The polar coordinate system uses distances and directions to specify the location of a point in the plane.

For this system, choose a fixed point O in the plane called the pole (or origin) and draw from O a ray (half-line) called the polar axis.

Then each point P can be assigned polar coordinates $P\left(r,\theta \right)$ where

r is the distance from O to P and

  $\theta$ is the angle between the polar axis and the segment $\overline{OP}$ .

  

The convention, that $\theta$ is positive if measured in the counter clockwise direction from the polar axis or negative if measured in the clockwise direction.

If r is negative, then $P\left(r,\theta \right)$ is defined to be the point that lies $\left|r\right|$ units from the pole in the direction opposite to that given by $\theta$ .

Conclusion:

Therefore, $P\left(r,\theta \right)$ is defined to be the point that lies $\left|r\right|$ units from the pole in the direction opposite to that given by $\theta$ .