Referred to the origin $O$, let $P$, $Q$ and $R$ be distinct points with position vecto

Question

Referred to the origin $O$, let $P$, $Q$ and $R$ be distinct points with position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. Show that $\left( \mathbf{r}-\mathbf{p} \right)\times \left( \mathbf{r}-\mathbf{q} \right)=\mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p}$.[2] Give the geometrical meaning of $\frac{1}{2}\left| \mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p} \right|$.[2] Given that $\mathbf{p}\times \mathbf{q}+\mathbf{q}\times \mathbf{r}+\mathbf{r}\times \mathbf{p}=\mathbf{0}$, $PR=3QR$, and that $PQ>PR$, express $\mathbf{r}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.[3]

Accepted Answer

(a) $(\mathbf r-\mathbf p)\times(\mathbf r-\mathbf q)=\mathbf p\times\mathbf q+\mathbf q\times\mathbf r+\mathbf r\times\mathbf p$ (b) The area of $\triangle PQR$ (c) $\mathbf r=\dfrac{\mathbf p+3\mathbf q}{4}$

2023 EJC Promo Q4

2023 EJC Promo Q4

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