Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and \(\mat

Question

Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $C$ lies on $OA$, between $O$ and $A$, such that $OC : CA = 3 : 2$. Point $D$ lies on $OB$, between $O$ and $B$, such that $OD : DB = 5 : 6$. Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$. Show that the vector equation of the line $BC$ can be written as $\mathbf{r} = \frac{3}{5} \lambda \mathbf{a} + (1 - \lambda) \mathbf{b}$, where $\lambda$ is a parameter. Find in similar form the vector equation of the line $AD$ in terms of a parameter $\mu$. Find, in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of the point $E$ where the lines $BC$ and $AD$ meet and find the ratio $AE : ED$.

Accepted Answer

(i) $\overrightarrow{OC} = \frac{3}{5}\mathbf{a}, \overrightarrow{OD} = \frac{5}{11}\mathbf{b}$ (ii) $l_{AD} : \mathbf{r} = \frac{5}{11}\mu\mathbf{b} + (1 - \mu)\mathbf{a}, \mu \in \mathbb{R}$ (iii) $\overrightarrow{OE} = \frac{9}{20}\mathbf{a} + \frac{1}{4}\mathbf{b}, AE : ED = 11 : 9$

N2015 P1 Q7

N2015 P1 Q7

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