Question

PARAGRAPH 1

Let $O$ be the origin, and $\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}$ be three unit vectors in the directions of the sides $\overrightarrow{Q R}, \overrightarrow{R P}, \overrightarrow{P Q}$ respectively, of a triangle $PQR$

If the triangle $P Q R$ 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)

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