Mathematics - Vector Question with Solution | TestHub

$\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ are unit vectors in the directions of sides $\overrightarrow{OR},\overrightarrow{RP}$ and $\overrightarrow{PQ}$ respectively,

$\therefore \overrightarrow{OX}=\frac{\overrightarrow{QR}}{\left|\overrightarrow{QR}\right|},\overrightarrow{OY}=\frac{\overrightarrow{RP}}{\left|\overrightarrow{RP}\right|},\overrightarrow{OZ}=\frac{\overrightarrow{PQ}}{\left|\overrightarrow{PQ}\right|}$

$\therefore \left|\overrightarrow{OX}\times \overrightarrow{OY}\right|=\frac{\left|\overrightarrow{OR}\times \overrightarrow{RP}\right|}{\left|\overrightarrow{QR}\right|\left|\overrightarrow{RP}\right|}$

$=\frac{\left|\overrightarrow{QR}\right|\left|\overrightarrow{RP}\right|\sin \left(P+Q\right)}{\left|\overrightarrow{QR}\right|\left|\overrightarrow{RP}\right|}$

$=\sin \left(P+Q\right)$