Paragraph: Let O be the origin, and O X , O Y , O Z be three unit vectors in the directions of the sides Q R , R P , P Q , respectively, of a triangle P Q R . Question: If the triangle P Q R varies, then the minimum value of (P+Q)+ (Q+R)+ (R+P) is

Step-by-step solution

cos P + Q + cos Q + R + cos R + P = cos 180 − R + cos 180 − P + cos 180 − Q = − cos P + cos Q + cos R In any Δ P Q R , cos P + cos Q + cos R ≤ 3 2 ⇒    − cos P + cos Q + cos R ≥ − 3 2