1.7: Unit vectors, constants and collinear points

0606/13/M/J/20 — Question 6 · 8 marks

(a)

Find the unit vector in the direction of

\begin{pmatrix} 5 \\ -12 \end{pmatrix}

[1]

(b)

Given that

\begin{pmatrix} 4 \\ 1 \end{pmatrix} + k\begin{pmatrix} -2 \\ 3 \end{pmatrix} = r\begin{pmatrix} -10 \\ 5 \end{pmatrix}

, find the value of each of the constants

k

and

r

[3]

(c)(i)

Relative to an origin

O

, the points

A

B

and

C

have position vectors

\mathbf{p}

3\mathbf{q} - \mathbf{p}

and

9\mathbf{q} - 5\mathbf{p}

respectively.

Find

\overrightarrow{AB}

in terms of

\mathbf{p}

and

\mathbf{q}

[1]

(c)(ii)

Find

\overrightarrow{AC}

in terms of

\mathbf{p}

and

\mathbf{q}

[1]

(c)(iii)

Explain why

A

B

and

C

all lie in a straight line.

[1]

(c)(iv)

Find the ratio

AB:BC

[1]