PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and \(\overrightarrow {{\text{QR}}}  = \) \(\boldsymbol{b}\),

(a)     express the vectors \(\overrightarrow {{\text{PR}}} \) and \(\overrightarrow {{\text{QS}}} \) in terms of \(\boldsymbol{a}\) and \(\boldsymbol{b}\);

(b)     hence show that the diagonals in a rhombus intersect at right angles.

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

\(\overrightarrow {{\text{QS}}}  = \) b − a     A1

[2 marks]

(b)     \(\overrightarrow {{\text{PR}}}  \cdot \overrightarrow {{\text{QS}}}  = \) (a + b) \( \cdot \) (b − a)     M1

\( = |\)b\({|^2} - |\)a\({|^2}\)     A1

for a rhombus \(|\)a\(| = |\)b\(|\)     R1

hence \(|\)b\({|^2} - |\)a\({|^2} = 0\)     A1

Note:     Do not award the final A1 unless R1 is awarded.

hence the diagonals intersect at right angles     AG

[4 marks]

Total [6 marks]