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In a quadrilateral P Q R S , vec P Q= ve...

In a quadrilateral `P Q R S , vec P Q= vec a , vec Q R = vec b , vec S P= vec a- vec b ,M` is the midpoint of ` vec Q Ra n dX` is a point on `S M` such that `S X=4/5S Mdot` Prove that `P ,Xa n dR` are collinear.

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`vec(OM) = (vecb)/(2) rArr vec(PM) = veca + (vecb)/(2)`

`vec(SM) = vec(PM) -vec(PS) = 2veca-(1)/(2) vecb`
`vec(SX) = (4)/(5) vec(SM) = (8)/(5) veca - (2)/(5) vecb`
`vec(PX)= vec(PS) + vec(SX)`
`" "=-veca+vecb+ (8)/(5) veca- (2)/(5) vecb= (3)/(5) (veca+ vecb)`
Also `vec(PR) = vec(PQ) + vec(QR) = veca+ vecb= (5)/(3) vec(PX)`
Hence P, X and R are collinear.
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