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

Question

Referred to the origin $O$, the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ where $\mathbf{a}$ and $\mathbf{b}$ are non-parallel vectors. It is also given that $C$, $D$ and $E$ are the midpoints of $OA$, $OB$ and $AB$ respectively, and $M$ is the point of intersection of the lines $AD$ and $BC$. Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Show that $M$ lies on the line $OE$. Find the area of the quadrilateral $OCMD$, expressing your answer in the form $k\left| \mathbf{a}	imes \mathbf{b} ight|$, where $k$ is a constant to be determined.

Accepted Answer

(i) $\overrightarrow{OM}=\frac13(\mathbf a+\mathbf b)$. (ii) $M$ lies on $OE$. (iii) $k=\frac16$.

NJC Vectors Tutorial Q6

NJC Vectors Tutorial Q6

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