In the diagram, OA = 2p, OB = 3q and OC = 4p + 9q.
(i) Given that the point P is such that AP = 2 PB, express the position vector of P in terms of p and q.
(ii) Given that point Q is such that OQ = 3 OP, express OQ in terms of p and q. Show that Q lies on BC and write down the numerical value of BQ/QC.
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Answer:
(i) AP = 2 PB
AO + OP = 2 (PO + OB)
OP - OA = -2 OP + 2 OB
3 OP = OA + 2 OB
3 OP = 2p + 6q
OP = ⅔p + 2q
(ii) OQ = 3 OP = 2p + 6q
BC = BO + OC
= -3q + 4p + 9q
= 4p + 6q
BQ = BO + OQ
= -3q + 2p + 6q
= 2p + 3q
Since
BQ = 2p + 3q
= ½ (4p + 6q)
= ½ BC
BQ and BC are collinear.
Hence, Q lies on BC
BQ/BC = ½
Thus, BQ / QC = 1
