The points $P$, $Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and \(\mathbf

Question

The points $P$, $Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. $P$ and $Q$ are fixed points and $R$ is a variable point. Describe geometrically the set of possible positions of $R$ such that $\mathbf{r} \cdot \mathbf{q} = \mathbf{p} \cdot \mathbf{q}$[1] $\mathbf{r} \times \mathbf{q} = \mathbf{p} \times \mathbf{q}$[2] With respect to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. $C$ lies on $AB$ produced such that $AC : CB = 3 : 2$. Given that $\mathbf{a}$ and $\mathbf{b}$ are unit vectors, the length of $OC$ is $\sqrt{7}$ units and the angle $AOB$ is $60^\circ$, find $|2\mathbf{a} - \mathbf{c}|$ by using a suitable scalar product.[5]

Accepted Answer

(a)(i) plane through $P$ with normal vector $\mathbf q$ (ii) line through $P$ parallel to $\mathbf q$ (b) $\sqrt{13}$

2023 JPJC P2 Q5

2023 JPJC P2 Q5

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