2.10: Intersection point expressed two ways

0606/23/O/N/17 — Question 5 · 8 marks

2.10 diagram

The diagram shows points

O

,

A

,

B

,

C

,

D

and

X

. The position vectors of

A

,

B

and

C

relative to

O

are

\overrightarrow{OA} = \mathbf{a}

,

\overrightarrow{OB} = \mathbf{b}

and

\overrightarrow{OC} = \frac{3}{2}\mathbf{b}

. The vector

\overrightarrow{CD} = 3\mathbf{a}

.

(i)

If

\overrightarrow{OX} = \lambda\overrightarrow{OD}

express

\overrightarrow{OX}

in terms of

\lambda

,

\mathbf{a}

and

\mathbf{b}

.

[1]

(ii)

If

\overrightarrow{AX} = \mu\overrightarrow{AB}

express

\overrightarrow{OX}

in terms of

\mu

,

\mathbf{a}

and

\mathbf{b}

.

[2]

(iii)

Use your two expressions for

\overrightarrow{OX}

to find the value of

\lambda

and of

\mu

.

[3]

(iv)

Find the ratio

\dfrac{AX}{XB}

.

[1]

(v)

Find the ratio

\dfrac{OX}{XD}

.

[1]