Three distinct non-zero vectors are given by $\overrightarrow {{\text{OA}}}$ = a, $\overrightarrow {{\text{OB}}}$ = b, and $\overrightarrow {{\text{OC}}}$ = c .

If $\overrightarrow {{\text{OA}}}$ is perpendicular to $\overrightarrow {{\text{BC}}}$ and $\overrightarrow {{\text{OB}}}$ is perpendicular to $\overrightarrow {{\text{CA}}}$ , show that $\overrightarrow {{\text{OC}}}$ is perpendicular to $\overrightarrow {{\text{AB}}}$.

$\overrightarrow {{\text{BC}}}$ = c – b

$\overrightarrow {{\text{CA}}}$ = a – c

$\Rightarrow$ a$\cdot$(c – b) = 0     M1

and b$\cdot$(a – c) = 0     M1

$\Rightarrow$ a$\cdot$c = a$\cdot$b     A1

and a$\cdot$b = b$\cdot$c     A1

$\Rightarrow$ a$\cdot$c = b$\cdot$c     M1

$\Rightarrow$ b$\cdot$c – a$\cdot$c = 0

c$\cdot$(b – a) = 0     A1

$\Rightarrow$ $\overrightarrow {{\text{OC}}}$ is perpendicular to $\overrightarrow {{\text{AB}}}$ , as b $\ne$ a .     AG

[6 marks]

Only the better candidates were able to make significant progress with this question. Many candidates understood how to begin the question, but did not see how to progress to the last stage. On the whole the candidates' use of notation in this question was poor.