2.7: Collinear points and a unit vector

0606/23/M/J/16 — Question 7 · 7 marks

O

P

Q

and

R

are four points such that

\overrightarrow{OP} = \mathbf{p}

\overrightarrow{OQ} = \mathbf{q}

and

\overrightarrow{OR} = 3\mathbf{q} - 2\mathbf{p}

(i)(a)

Find, in terms of

\mathbf{p}

and

\mathbf{q}

, the vector

\overrightarrow{PQ}

[1]

(i)(b)

Find, in terms of

\mathbf{p}

and

\mathbf{q}

, the vector

\overrightarrow{QR}

[1]

(ii)

Justifying your answer, what can be said about the positions of the points

P

Q

and

R

[2]

(iii)

Given that

\overrightarrow{OP} = \mathbf{i} + 3\mathbf{j}

and that

\overrightarrow{OQ} = 2\mathbf{i} + \mathbf{j}

, find the unit vector in the direction

\overrightarrow{OR}

[3]