In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vec

Question

In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively. State a geometrical interpretation of $|\mathbf{c} \times \mathbf{\hat{a}}|$ and hence show that the area of the triangle $PQR$ can be written as $\frac{1}{2}|\mathbf{c} \times \mathbf{{a}}|$ units$^2$.[3] With clear explanations, show that $|\mathbf{c} \times \mathbf{a}| = |\mathbf{b} \times \mathbf{a}|$ and deduce that $\frac{|\mathbf{c}|}{\sin \angle PQR} = \frac{|\mathbf{b}|}{\sin \angle QPR}$ (Sine Rule).[2]

Accepted Answer

(a) ${\frac{1}{2}\left| \mathbf{c}	imes \mathbf{a} ight|	ext{ units}^2 	ext{(Shown)}}$ (b) ${\left| \mathbf{c}	imes \mathbf{a} ight| = \left| \mathbf{b}	imes \mathbf{a} ight| \Rightarrow \left| \mathbf{c} ight|\left| \mathbf{a} ight|\sin \angle QPR = \left| \mathbf{b} ight|\left| \mathbf{a} ight|\sin \angle PQR}$ ${\Rightarrow \frac{\left| \mathbf{c} ight|}{\sin \angle PQR} = \frac{\left| \mathbf{b} ight|}{\sin \angle QPR}	ext{(Shown)}}$

2024 ASRJC Promo Q5

2024 ASRJC Promo Q5