Adding Vectors (r, θ, φ) - A Step-by-Step Guide

[Philosophaie]

How do you add two vectors:

(r1, theta1, phi1) + (r2, theta2, phi2)

I know if you break them down into x, y, z components you get the right answer but how do you add these vectors.

 

[Erland]

As you said, transform them to rectangular coordinates (xyz-components), add them, and then transform back to spherical coordinates.

 

[HallsofIvy]

If you mean "without using components" (i.e. the hard way) you use trigonometry.

Draw a line segment with one end at (0, 0, 0) in the direction given by theta1 and phi1 with length r1. At its tip, draw a second line segment in the direction given by theta2 and phi2 with length r2. Finally, draw the line segment from (0, 0, 0) to the tip of the second line segment. This forms a triangle in which you know the length of two sides and the angle between them so can use the sine and cosine laws to determine the length and angles of the third side. It is determining the angle between the two sides that is complicated.