Write down expressions for \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{CB}}} \) in terms of the vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\) .

Hence prove that angle \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle.

\(\overrightarrow {{\text{AB}}}  = {\boldsymbol{b}} - {\boldsymbol{a}}\)     A1

\(\overrightarrow {{\text{CB}}}  = {\boldsymbol{a}} + {\boldsymbol{b}}\)     A1

[2 marks]

\(\overrightarrow {{\text{AB}}}  \cdot \overrightarrow {{\text{CB}}}  = \left( {{\boldsymbol{b}} - {\boldsymbol{a}}} \right) \cdot \left( {{\boldsymbol{b}} + {\boldsymbol{a}}} \right)\)     M1
\( = {\left| {\mathbf{b}} \right|^2} - {\left| {\mathbf{a}} \right|^2}\)     A1
\( = 0\) since \(\left| {\boldsymbol{b}} \right| = \left| {\boldsymbol{a}} \right|\)     R1

Note: Only award the A1 and R1 if working indicates that they understand that they are working with vectors.

so \(\overrightarrow {{\text{AB}}} \) is perpendicular to \(\overrightarrow {{\text{CB}}} \) i.e. \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle     AG

[3 marks]

This question was poorly done with most candidates having difficulties in using appropriate notation which made unclear the distinction between scalars and vectors. A few candidates scored at least one of the marks in (a) but most candidates had problems in setting up the proof required in (b) with many using a circular argument which resulted in a very poor performance in this part.

This question was poorly done with most candidates having difficulties in using appropriate notation which made unclear the distinction between scalars and vectors. A few candidates scored at least one of the marks in (a) but most candidates had problems in setting up the proof required in (b) with many using a circular argument which resulted in a very poor performance in this part.