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) $|\mathbf c\times\hat{\mathbf a}|$ is the perpendicular distance from $R$ to $PQ$, and the area is $\dfrac12|\mathbf c\times\mathbf a|\text{ units}^2$ (b) $|\mathbf c\times\mathbf a|=|\mathbf b\times\mathbf a|$, hence $\dfrac{|\mathbf c|}{\sin\angle PQR}=\dfrac{|\mathbf b|}{\sin\angle QPR}$

2024 ASRJC Promo Q5

2024 ASRJC Promo Q5

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