Polar Coordinates Another way of specifying a vector is to give its length and the angle it encloses with the axes of the coordinate system. This can be done in two and in three dimensions, but gets unwielding pretty quickly in the latter. So we will limit ourselves to two dimensions. In two dimensions, the angle used is the one enclosed with the x-axis. So you could say a vector is given by "length 5 meters, angle 30 degrees." Polar to Coordinate Representation: To get back to the coordinate representation, you would use a = a cos a sin with a being the length ("5 meters") and being the angle ("30 degrees"). As you look at the drawing, you can see that this is just straightforward trigonometry. That a is indeed the length of this vector, you can see from || = a sin + a cos 0 = a sin 0 + cos 0 = a using the trigonometric identity that sin + cos = 1 for any angle 0. Coordinate to Polar Representation: Given the cartesian coordinates, az a = ay it is a bit more involved to get the polar representation, since the mechanism depends on the quadrant you are in: a 0:0 = 90 arctan(a/an) . az > 0; ay > 0:0 = arctan(ay/ar) . az 0;ay