Referred to the origin $O$, the points $A$ and $B$ are such that \(\overrightarrow{OA}=\m

Question

Referred to the origin $O$, the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. The point $C$ lies on $OB$ such that $\overrightarrow{OC}=p\overrightarrow{OB}$, where $p$ is a constant. $D$ is on $AC$ such that $AD:DC=5:3$ and $E$ is on $AB$ such that $AE:EB=3:2$. Find $\overrightarrow{OD}$ and $\overrightarrow{OE}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $p$.[2] Given that $O$, $D$ and $E$ are collinear, find $p$.[3] Show that the area of the triangle $OAE$ can be expressed as $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be determined.[3]

Accepted Answer

(i) $\overrightarrow{OE}=\frac{2\mathbf{a}+3\mathbf{b}}{5}$ (ii) $\frac{9}{10}$ (iii) $k=\frac{3}{10}$

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