Let P be the point #(-6, pi/3)# , which is #(r, theta)# in polar coordinates, and Q be the point #(7, pi/2)# like wise in the polar coordinates. To plot the point P, move along the ray making an angle #theta=pi/3# with positive x-axis at the origin and extend the ray backwards in the IIIrd quadrant. Since #r# is negative, length 6 units would be measured along the ray in the third quadrant. As shown in the calculation below the cartesean coordinates of P would be #(-3, -3sqrt3)#.

Similarly the cartesean coordinates of Q would be (0,7).

The distance PQ would be #sqrt ((3-0)^2 +(3sqrt3 -7)^2)#

= #sqrt (9+27 +49 -42sqrt3)#

=#sqrt (85-42sqrt3)#