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Let O be the origin and vec(OX) , vec(O...

Let O be the origin and` vec(OX) , vec(OY) , vec(OZ)` be three unit vector in the directions of the sides `vec(QR) , vec(RP),vec(PQ)` respectively , of a triangle PQR.
if the triangle PQR varies , then the manimum value of `cos (P+Q) + cos(Q+R)+ cos (R+P)` is

A

` - 5/3`

B

` -3/2`

C

`3/2`

D

`5/2`

Text Solution

Verified by Experts

The correct Answer is:
B
In ` trianglePQR, P +Q +R= pi`
we have, ` OvecX, OvecY = cos ( pi-R) =-cos R`
` OvecY. OvecZ = cos ( pi- R) =- cos P`
` OvecZ, OvecX = cos ( pi -Q) =- cos Q`

` cos ( P +Q) +cos ( Q +R) +cos ( R +P)`
` = cos ( pi- R) + cos ( pi -P) ( + cos ( pi- Q) `
` = - ( cos P + cos Q + cos R)`
`= OvecX. OvecY+ OvecY.OvecZ+ vec(OZ).vec(OX)`
`= 1/2 [|vec(OX)+vec(OY)+vec(OZ)|^(2)-{|vec(OX)|^(2)+|vec(OY)|^(2) + |vec(OZ)|^(2)}]`
`=1/2[|vec(OX)+vec(OY)+vec(OZ)|^(2)-3]=1/2{|vec(OX)|^(2)+|vec(OY)|^(2) +|vec(OZ)|^(2)|^(2)} -3/2`
`ge -3/2 " " [because1/2{|vec(OX)|^(2)+|vec(OY)|^(2) +|vec(OZ)|^(2)} ge0]`
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