P,Q,R,S are four vectors of equal magnitude. P + Q - R = 0, where the angle between P and Q is theta_1. P + Q - S = 0, where the angle between P and S is theta_2. What is the ratio of theta_1 and theta_2?

Hello!
First, rewrite the equalities as P + Q = R and P + Q = S. We see now that R = S, so theta_1 is the angle between P and Q, theta_2 is the angle between P and P + Q.
P + Q is the diagonal of the parallelogram formed by P and Q. Because P and Q have the equal magnitudes, this parallelogram is a rhombus, and its diagonals bisect the angles between the sides.
Thus theta_2 is a half of theta_1 , and the ratio is theta_1 : theta_2 is 2 : 1.
If we take into account that the magnitude of P+Q is the same of P and Q, we find that theta_2=pi/3 and theta_1=(2 pi)/3.