With respect to origin $O$, the distinct points $P$, $Q$, $R$ and $S$ have position vector

Question

With respect to origin $O$, the distinct points $P$, $Q$, $R$ and $S$ have position vectors $\mathbf{p}$, $\mathbf{q}$ , $\mathbf{r}$ and $\mathbf{s}$ respectively. It is known that $\mathbf{q}$ is a unit vector. Given that $\left( \mathbf{r}-\mathbf{p} \right)\times \left( \mathbf{p}-\mathbf{q} \right)=\mathbf{0}$, state, with justification, the relationship between the points $P$, $Q$ and $R$.[2] Give a geometrical interpretation of $\left| \mathbf{q}\cdot \left( \mathbf{r}-\mathbf{s} \right) \right|$.[1] Given that $\mathbf{p}=\left( \begin{matrix}-1 \2 \4 \\end{matrix} \right)$ and that $\mathbf{q}$ is parallel to and in the opposite direction of $\mathbf{p}$, find $\mathbf{q}$.[2]

Accepted Answer

(i) $R$ is a point on line passing through $P$ and $Q$. (ii) $\left| \mathbf{q} \cdot (\mathbf{r} - \mathbf{s}) \right|$ is the length of projection of $\overrightarrow{SR}$ onto $\mathbf{q}$. (iii) $-\frac{1}{\sqrt{21}} \begin{pmatrix} -1 \ 2 \ 4 \end{pmatrix}$

2021 HCI Promo Q4

2021 HCI Promo Q4

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