Three distinct points $O$, $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrigh

Question

Three distinct points $O$, $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Let $\overrightarrow{OP}=\mathbf{r}$ and $\mathbf{r}	imes \mathbf{b}=\mathbf{a}	imes \mathbf{b}$. By expressing $\mathbf{r}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, show that the locus of $P$ is a line parallel to $\mathbf{b}$. [2] In terms of $\mathbf{a}$ and $\mathbf{b}$, write down the distance from $O$ to this line. Hence write down the distance from $B$ to this line. [2] Given that the line passes through $O$, what can you say about the relationship between $\mathbf{a}$ and $\mathbf{b}$? [1]

Accepted Answer

(a)(i) Locus of $P$ is all points on the line parallel to $\mathbf{b}$ and passes through point $A$ (a)(ii) Distance of $O$ to line $= \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{b}|}$ (a)(iii) $\mathbf{a}$ is parallel to $\mathbf{b}$

2016 HCI J2 CT P1 Q10

2016 HCI J2 CT P1 Q10

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