Given that P=Q=R. If P+Q=R, then the angle between P and R is ${\theta }_1$. If P+Q+R=0, then the angle between P and R is ${\theta }_2$. What is the relation between ${\theta }_1$ and ${\theta }_2$?

$\mathbf{P}+\mathbf{Q}=\mathbf{R}$ and P=Q=R

$R=\sqrt{P^2+Q^2+2PQ\cos \theta }$

Substituting P=Q=R, we get $\theta =120°$

If P=Q, then R will pass through the centre of P and Q. Hence, angle between P and R will be

${\theta }_1=\frac{\theta }{2}=60°$

In the second condition, given that P=Q=R and $\mathbf{P}+\mathbf{Q}+\mathbf{R}=0$. Then, angle between P and R will be ${\theta }_2=120°$

${\theta }_1=\frac{{\theta }_2}{2}$