In triangle $OAB$, $C$ is on $OA$, $D$ is on $AB$, such that $BC$ intersects $OD$

Question

In triangle $OAB$, $C$ is on $OA$, $D$ is on $AB$, such that $BC$ intersects $OD$ at $E$. It is given that $\overrightarrow{OE} = \frac{1}{2}\overrightarrow{OD}$, $\overrightarrow{EB} = \frac{2}{3}\overrightarrow{CB}$, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Express $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Given that $\overrightarrow{AD} = \lambda \overrightarrow{AB}$, express $\overrightarrow{OE}$ in terms of $\lambda$, $\mathbf{a}$ and $\mathbf{b}$. Given that $\overrightarrow{OC} = \mu \overrightarrow{OA}$, express $\overrightarrow{OE}$ in terms of $\mu$, $\mathbf{a}$ and $\mathbf{b}$. Find the values of $\lambda$ and $\mu$.

Accepted Answer

(a) $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}$ (b)(i) $=\frac{1-\lambda }{2}\mathbf{a}+\frac{\lambda }{2}\mathbf{b}$ (ii) $=\frac{2}{3}\mu \mathbf{a}+\frac{1}{3}\mathbf{b}$ (c) $\lambda =\frac{2}{3}$ $\mu =\frac{1}{4}$

Vectors - Vector Subtraction

Vectors - Vector Subtraction