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This fact is easy, although rather tedious, to demonstrate rigorously (In pure mathematics, the term rigor refers to the process of proving something in a series of absolutely logical steps It has nothing to do with the physical condition called rigor mortis) We must define the two vectors by coordinates, and then work through the arithmetic with those coordinates Let s call the two vectors a = (xa,ya) and b = (xb,yb) As defined earlier in this chapter, the Cartesian sum a + b is a + b = [(xa + xb),(ya + yb)] Using the same definition, the Cartesian sum b + a is b + a = [(xb + xa),(yb + ya)] All four of the coordinate values xa, xb, ya, and yb are real numbers We know from basic algebra that addition of real numbers is commutative Therefore, we can reverse both of the sums in the elements of the ordered pair above, getting b + a = [(xa + xb),(ya + yb)] That s the ordered pair that defines a + b We have just shown that a+b=b+a for any two Cartesian vectors a and b

In the polar coordinate plane, we draw a vector as a ray going straight outward from the origin to a point defined by a specific angle and a specific radius Figure 4-6 shows two vectors a and b with originating points at (0,0) and terminating points at (qa,ra) and (qb,rb), respectively

Polar magnitude and direction The magnitude and direction of a vector a = (qa,ra) in the polar coordinate plane are defined directly by the coordinates The magnitude is ra, the straight-line distance of the terminating point from the origin The direction angle is qa, the angle that the ray subtends in a

(q b, rb)

p /2

(qa, ra)

3p /2

ordered pairs for their terminating points, denoting the direction angle (relative to the reference axis marked 0) and the radius (the distance from the origin)

counterclockwise sense from the reference axis (labeled 0 here) By convention, we restrict the vector magnitude and direction to the ranges ra 0 and 0 qa < 2p If a vector s magnitude is 0, then the direction angle doesn t matter; the usual custom is to set it equal to 0

Special constraints When defining polar vectors, we must be more particular about what s legal and what s illegal than we were when defining polar points in Chap 3 With polar vectors:

We don t allow negative magnitudes We don t allow negative direction angles We don t allow direction angles of 2p or larger These constraints ensure that the set of all polar-plane vectors can be paired off in a one-to-one correspondence (also called a bijection) with the set of all Cartesian-plane vectors

Polar vector sum If we have two vectors in polar form, their sum can be found by following these steps, in order:

1 Convert both vectors to Cartesian coordinates 2 Add the vectors the Cartesian way 3 Convert the Cartesian vector sum back to polar coordinates Let s look at the situation in more formal terms Suppose we have two vectors expressed in polar form as a = (qa,ra) and b = (qb,rb) To convert these vectors to Cartesian coordinates, we can use formulas adapted from the polar-to-Cartesian conversion we learned in Chap 3 The modified formulas are (xa,ya) = [(ra cos qa),(ra sin qa)] and (xb,yb) = [(rb cos qb),(rb sin qb)] Once we have obtained the Cartesian ordered pairs, we add their elements individually to get a + b = [(xa + xb),(ya + yb)] Let s call this Cartesian sum vector c, and say that c = a + b = [(xa + xb),(ya + yb)] = (xc,yc) To convert c from Cartesian coordinates into polar coordinates, we can use the formulas given earlier in this chapter for the magnitude and direction angle of a vector in the xy plane If we call the magnitude rc and the direction angle qc, we can write down the polar coordinates of sum vector as c = (qc, rc)