iii)

The position vector of

D

is

O D

.
Identify a path from

O

to

D

that uses some of the paths you already know vectors for.

\begin{array}{rcl} \vec{O D} & = & \vec{O E} + \vec{E D} \end{array}

E

is the midpoint of

A C

so

\begin{array}{rcl} \vec{E C} & = & \frac{1}{2} \vec{A C} = \frac{1}{2} a \end{array}

. Therefore

\begin{array}{rcl} \vec{E D} \end{array}

must be equal to

\begin{array}{rcl} \frac{1}{2} a + x b \end{array}

where

\begin{array}{rcl} x \end{array}

is a constant to be found.

\begin{array}{rcl} \vec{E D} \end{array}

is parallel to

\begin{array}{rcl} \vec{O E} \end{array}

so

\begin{array}{rcl} \vec{E D} \end{array} = k (\begin{array}{rcl} \vec{O E} \end{array}) = k (\frac{3}{2} a + b)

.

\begin{array}{rcl} \frac{1}{2} a + x b & = & k (\frac{3}{2} a + b) \end{array}

Compare coefficients of a.

\begin{array}{rcl} \frac{1}{2} a & = & k (\frac{3}{2} a) \\ \frac{3}{2} k & = & \frac{1}{2} \\ k & = & \frac{1}{3} \end{array}

Compare coefficients of b.

x b = k b = \frac{1}{3} b

so

\begin{array}{rcl} \vec{E D} & = & \frac{1}{2} a + \frac{1}{3} b \end{array}

.

[1]

Substitute these into the route for the path.

\begin{array}{rcl} \vec{O D} & = & \vec{O E} + \vec{E D} = (\frac{3}{2} a + b) + (\frac{1}{2} a + \frac{1}{3} b) \end{array}

[1]

Simplify.

\begin{array}{rcl} \vec{O D} & = & (\frac{3}{2} a + \frac{1}{2} a) + (b + \frac{1}{3} b) \end{array}

\vec{O D} = 2 a + \frac{4}{3} b

[1]

Vectors (Cambridge O Level Maths)

Exam Questions