Plotting a Point in the Polar Coordinate System with a Negative Component

We know that $\frac{\pi }{6}$ is located in the first quadrant. However, $r=\mathrm{-2.}$ We can approach plotting a point with a negative $r$ in two ways:

1. Plot the point $\left(2,\frac{\pi }{6}\right)$ by moving $\frac{\pi }{6}$ in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
2. Move $\frac{\pi }{6}$ in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See Figure 4(a). Compare this to the graph of the polar coordinate $\left(2,\frac{\pi }{6}\right)$ shown in Figure 4(b).

Figure 4