Question

Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ},$ respectively, of a triangle $PQR.$

If the triangle $PQR$ varies, then the minimum value of
$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$

• A. $-\frac{5}{3}$
• B. $-\frac{3}{2}$
• C. $\frac{3}{2}$
• D. $\frac{5}{3}$

Answer 1

The correct option is B $-\frac{3}{2}$


$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$
$=-\cos R-\cos P-\cos Q=-(\cos R+\cos P+\cos Q)$
In any triangle, the maximum value of $\cos P+\cos Q+\cos R=\frac{3}{2}$
$\therefore$ Minimum value of the given expression $\cos (P+Q)+\cos (Q+R)+\cos (R+P)=-\frac{3}{2}$