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Question
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are ^i+2^j−^k and −^i+^j+^k respectively, in the ratio 2:1
externally.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are ^i+2^j−^k and −^i+^j+^k respectively, in the ratio 2:1
externally.
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Solution
The position vector of a point R divided the line segment joining two points P and Q in the ratio m : n is given by
Case I Internally= mb+nam+n
Case II Externally=mb−nam−n
Position vectors of P and Q are given as
OP=^i+2^j−^k and OQ=−^i+^j+^k
PV of R [dividing (PQ) in the ratio 2 : 1 externally]
=m(PV of Q)−n(PV of P)m−n, here m=2, n=1=2(PV of Q)−1(PV of P)2−1
=2(−^i+^j+^k)+(−1)(^i+2^j−^k)=(−3)^i+0^j+3^k=−3^i+3^k
The position vector of a point R divided the line segment joining two points P and Q in the ratio m : n is given by
Case I Internally= mb+nam+n
Case II Externally=mb−nam−n
Position vectors of P and Q are given as
OP=^i+2^j−^k and OQ=−^i+^j+^k
PV of R [dividing (PQ) in the ratio 2 : 1 externally]
=m(PV of Q)−n(PV of P)m−n, here m=2, n=1=2(PV of Q)−1(PV of P)2−1
=2(−^i+^j+^k)+(−1)(^i+2^j−^k)=(−3)^i+0^j+3^k=−3^i+3^k
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respectively, in the ration 2:1