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Question

Show that the positive vector of the point P, which divides the line joining the points A and B having position vectors a and b internally in ratio m:n is mb+nam+n.

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Solution

Let O be the origin representing the position vectors OA and OB with respect to the two points A and B. Let A and B be divided by a third point P internally in the ratio m:n
APAB=mn or nAP=mPB
nAP=mPB
n(OPOA)=m(OBOP)
n(ra)=m(br)
nrna=mbmr
mr+nr=mb+na
or r=mb+nam+n (proved).
870745_946870_ans_0bd2f5b60806478398b93f8a3484b3e2.png

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