2.9: Triangle with cevians meeting at a point

0606/23/M/J/19 — Question 10 · 9 marks

The diagram shows a triangle

OAB

. The point

P

is the midpoint of

OA

and the point

Q

lies on

OB

such that

\overrightarrow{OQ} = \frac{1}{4}\overrightarrow{OB}

. The position vectors of

P

and

Q

relative to

O

are

\mathbf{p}

and

\mathbf{q}

respectively.

(i)

Find, in terms of

\mathbf{p}

and

\mathbf{q}

, an expression for each of the vectors

\overrightarrow{PQ}

\overrightarrow{QA}

and

\overrightarrow{PB}

[3]

(ii)

Given that

\overrightarrow{PR} = \lambda\overrightarrow{PB}

and that

\overrightarrow{QR} = \mu\overrightarrow{QA}

, find an expression for

\overrightarrow{PQ}

in terms of

\lambda

\mu

\mathbf{p}

and

\mathbf{q}

[2]

(iii)

Using your expressions for

\overrightarrow{PQ}

, find the value of

\lambda

and of

\mu

[4]