O, A, B and C are distinct points such that $\overrightarrow {{\text{OA}}}  =$ a, $\overrightarrow {{\text{OB}}}  =$ b and $\overrightarrow {{\text{OC}}}  =$ c.

It is given that c is perpendicular to $\overrightarrow {{\text{AB}}}$ and b is perpendicular to $\overrightarrow {{\text{AC}}}$.

Prove that a is perpendicular to $\overrightarrow {{\text{BC}}}$.

c $\bullet$ (b $-$ a) $= 0$     M1

Note:     Allow c $\bullet$ $\overrightarrow {{\text{AB}}}  = 0$ or similar for M1.

c $\bullet$ b $=$ c $\bullet$ a     A1

b $\bullet$ (c $-$ a) $= 0$

b $\bullet$ c $=$ b $\bullet$ a     A1

c $\bullet$ a $=$ b $\bullet$ a     M1

(c $-$ b) $\bullet$ a $= 0$     A1

hence a is perpendicular to $\overrightarrow {{\text{BC}}}$     AG

Note:     Only award the final A1 if a dot is used throughout to indicate scalar product.

Condone any lack of specific indication that the letters represent vectors.

[5 marks]

This was generally poorly done. The recent syllabus change refers to ‘proof of geometrical properties using vectors’ and this is clearly a topic candidates are not entirely clear with at the moment. Despite the question clearly being written as a vector question some students tried to use a geometrical approach, assuming it was two-dimensional. Many did not seem to realise that vectors being perpendicular implies that their scalar product is zero.