ACJC/2022/JC1/Promo/Q07

(a) Referred to the origin $O$, points $A$ and $B$ have position vectors a and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non parallel vectors. Point $C$ lies on $O A$, between $O$ and $A$, such that $O C: C A=2: 1$. Point $D$ lies on $O B$ produced such that $B D=5 O B$
Find, in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of the point $E$ where the lines $AB$ and $CD$ meet.

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(b) The position vectors of the points $D$ and $E$ relative to the origin $O$ are $\mathbf{d}$ and $\mathbf{e}$ respectively, where $\mathbf{d}$ and $\mathbf{e}$ are non-zero and non parallel vectors. It is given that the length $O E$ is 2 units and $|\mathbf{d} \cdot \mathbf{e}|=3$.

The point $F$, with position vector $\mathbf{f}$, is the reflection of the point $D$ in the line $O E$.
(i) Express $\mathbf{f}$ in terms of vectors $\mathbf{d}$ and $\mathbf{e}$.

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(ii) Show that the area of triangle $O D F$ can be expressed as $k|\mathbf{d} \times \mathbf{e}|$, where $k$ is a constant to be determined.

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