Two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively rela

Question

Two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively relative to origin $O$. The points $P$ on $OA$ and $Q$ on $AB$ are such that $OP=3PA$ and $5AQ=7QB$. The point $R$ is on $PQ$ produced so that $5PQ=4QR$. Express $\overrightarrow{AQ}$, $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Prove by a vector method that $O$, $B$, $R$ are collinear.[6]

Accepted Answer

$\overrightarrow{AQ}=\dfrac7{12}(\mathbf b-\mathbf a)$ $\overrightarrow{PQ}=\dfrac7{12}\mathbf b-\dfrac13\mathbf a$ $\overrightarrow{PR}=\dfrac{21}{16}\mathbf b-\dfrac34\mathbf a$ $\overrightarrow{OR}=\dfrac{21}{16}\overrightarrow{OB}$, so $O$, $B$, and $R$ are collinear

JPJC Vectors Tutorial 1(a) Q3

JPJC Vectors Tutorial 1(a) Q3

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