With reference to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$

Question

With reference to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. Point $C$ lies between $O$ and $A$ such that $OC:CA=3:2$. Point $X$ lies between $A$ and $B$ such that $AX:XB=1:3$. The lines $BC$ and $OX$ meet at the point $E$. Show that the vector equation of the line $BC$ can be written as $\mathbf{r}=\frac{3}{5}\lambda \mathbf{a}+\left( 1-\lambda \right)\mathbf{b}$ where $\lambda $ is a parameter.[1] By finding a vector equation of the line $OX$ and using the result in (a), show that the position vector of $E$ is $\frac{1}{2}\mathbf{a}+\frac{1}{6}\mathbf{b}$.[4] It is given that angle $AOB$ is ${{30}^{\circ }}$ and $\mathbf{b}$ is a unit vector. Find the exact perpendicular distance from $E$ to the line $OA$.[3]

Accepted Answer

(a) $\mathbf r=\dfrac35\lambda\mathbf a+(1-\lambda)\mathbf b$, $\lambda\in\mathbb R$ (b) $\overrightarrow{OE}=\dfrac12\mathbf a+\dfrac16\mathbf b$ (c) $\dfrac1{12}\text{ unit}$

2023 TJC Promo Q6

2023 TJC Promo Q6

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