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) $\mathbf b-\mathbf a$ (b)(i) $\dfrac{1-\lambda}{2}\mathbf a+\dfrac\lambda2\mathbf b$ (ii) $\dfrac{2\mu}{3}\mathbf a+\dfrac13\mathbf b$ (c) $\lambda=\dfrac23,\;\mu=\dfrac14$.

Vectors - Vector Subtraction

Vectors - Vector Subtraction

Solution:

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