Show that OC = 2AB .

Find the position vectors of C, D and E in terms of a and b .

${\text{OC}} = {\text{AB}} + {\text{OA}}\cos 60 + {\text{BC}}\cos 60$     M1

$= {\text{AB}} + {\text{AB}} \times \frac{1}{2} + {\text{AB}} \times \frac{1}{2}$     A1

$= 2{\text{AB}}$     AG

[2 marks]

$\overrightarrow {{\text{OC}}} = 2\overrightarrow {{\text{AB}}} =$2(b – a)     M1A1

$\overrightarrow {{\text{OD}}} = \overrightarrow {{\text{OC}}} + \overrightarrow {{\text{CD}}}$     M1

$= \overrightarrow {{\text{OC}}} + \overrightarrow {{\text{AO}}}$     A1

= 2b – 2a – a = 2b – 3a     A1

$\overrightarrow {{\text{OE}}} = \overrightarrow {{\text{BC}}}$     M1

= 2b – 2a – b = b – 2a     A1

[7 marks]