Now, $$\overrightarrow {OX} = {{\overrightarrow {QR} } \over {QR}}$$
and $$\overrightarrow {OY} = {{\overrightarrow {RP} } \over {RP}}$$
Therefore, $$(\overrightarrow {OX} \times \overrightarrow {OY} ) = {{\overrightarrow {QR} } \over {QR}} \times {{\overrightarrow {RP} } \over {RP}} = {{\overrightarrow {QR} \times \overrightarrow {RP} } \over {PQ}}$$
$$ = {{PQ\sin R} \over {PQ}} = \sin R = \sin (\pi - (P + Q) = \sin (P + Q))$$