An extended segment and coefficient comparison

Relative to origin O, P and Q have position vectors p,q\mathbf p,\mathbf q, nonparallel and nonzero. R lies on PQ produced with PQ:QR=1:λPQ:QR=1:\lambda. M is the midpoint of OR.

(i) Find the position vector of R in terms of λ,p,q\lambda,\mathbf p,\mathbf q. mark

F lies on OQ and F, P, M are collinear.

(ii) Find OF:FQOF:FQ in terms of λ\lambda. marks

[2018/EJC/Prelim/I/5]

Total marks: 5

Solution

(i)

Ratio theorem

RR lies on PQPQ produced with PQ:QR=1:λPQ:QR=1:\lambda. Find OR→\overrightarrow{OR}.

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Hint

  1. QQ divides PRPR in the ratio 1:λ1:\lambda (internally).

Answer · (i)

OR→=(1+λ)q−λp\overrightarrow{OR}=(1+\lambda)q-\lambda p.

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P, Q and R in a line (λ = 1)

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

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(ii)

Comparing coefficients

MM is the midpoint of OROR and FF lies on OQOQ with F,P,MF,P,M collinear. Find OF:FQOF:FQ in terms of λ\lambda.

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Hint

  1. OF→=tq\overrightarrow{OF}=t\mathbf q (on OQOQ).

  2. FF on PMPM: OF→=sOM→+(1−s)p\overrightarrow{OF}=s\overrightarrow{OM}+(1-s)\mathbf p.

  3. Equate and compare the coefficients of p\mathbf p and q\mathbf q.

Answer · (ii)

OF:FQ=(1+λ):1OF:FQ=(1+\lambda):1.

λ>0\lambda>0.

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F where PM meets OQ (λ = 1)

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.