Question

PARAGRAPH 1

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
$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$is

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

Answer 1

Answer: (B)

$-(\cos P+\cos Q+\cos R)=\overrightarrow{OX}\cdot \overrightarrow{OY}+\overrightarrow{OY}\cdot \overrightarrow{OZ}+\overrightarrow{OZ}\cdot \overrightarrow{OX}$
$=\frac{(\overrightarrow{OX}+\overrightarrow{OY}+\overrightarrow{OZ}{)}^2-\left(|\overrightarrow{OX}{|}^2+|\overrightarrow{OY}{|}^2+|\overrightarrow{OZ}{|}^2\right)}{2}$
$\ge -\frac{3}{2}$