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]