Question

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is

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

Answer 1

The correct option is A

$-\frac{3}{2}$

$\cos (P+Q)+\cos (Q+R)+\cos (R+P)=-(\cos R+\cos P+\cos Q)\text{Max. of}\cos P+\cos Q+\cos R=\frac{3}{2}\text{Min. of}\cos (P+Q)+\cos (Q+R)+\cos (R+P)=-\frac{3}{2}$