In triangle $OAB$, $O,A,B$ are not collinear, $\overrightarrow{OA}=\mathbf{a}$ and \(\overrigh

Question

In triangle $OAB$, $O,A,B$ are not collinear, $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Point $P$ is the midpoint of $OA$, and $Q$ lies on $OB$ with $OQ:QB=1:2$. The lines $AQ$ and $BP$ intersect at $R$. Let $\overrightarrow{AR}=t\overrightarrow{AQ}$ and $\overrightarrow{BR}=s\overrightarrow{BP}$. Express $\overrightarrow{OR}$ using the route through $A$, and then using the route through $B$.[2] Find $s$ and $t$.[3] Find $AR:RQ$.[1]

Accepted Answer

(a) $(1-t)\mathbf{a}+\dfrac t3\mathbf{b}$, $\dfrac s2\mathbf{a}+(1-s)\mathbf{b}$ (b) $s=\dfrac45,t=\dfrac35$ (c) $3:2$

Two routes to an intersection

Two routes to an intersection

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