Let $O$ be the origin, and $\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}$ and be

Question

Let $O$ be the origin, and $\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}$ and be three unit vectors in the directions of the sides $\overrightarrow{Q R}, \overrightarrow{R P}, \overrightarrow{P Q}$, respectively, of a triangle $P Q R$.\ If the triangle $P Q R$ varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is\

Professor PC Thomas Classes · Answer

(a) : $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$\ $=-\cos R-\cos P-\cos Q$\ $=-(\cos P+\cos Q+\cos R)$\ In a triangle, we know that\ $\cos P+\cos Q+\cos R \leq \frac{3}{2}$\ This is a standard result that gives\ $-(\cos P+\cos Q+\cos R) \geq-\frac{3}{2}$\ Thus the minimum value of\ $-(\cos P+\cos Q+\cos R)$ is $-3 / 2$.\