$O=(0,0)$, $A=(8,6)$ and $B=(-2,4)$. Let $\overrightarrow{OA}=\mathbf a$ and \(\overrig

Question

$O=(0,0)$, $A=(8,6)$ and $B=(-2,4)$. Let $\overrightarrow{OA}=\mathbf a$ and $\overrightarrow{OB}=\mathbf b$. Point $P$ divides $AB$ internally in the ratio $AP:PB=2:1$. Point $Q$ is defined by $\overrightarrow{OQ}=\dfrac12\mathbf a+\mathbf b$. The diagram is not drawn to scale; the apparent alignment is not a proof. Find $\overrightarrow{AB}$ as a column vector and calculate its magnitude.[2] Express $\overrightarrow{OP}$ in terms of $\mathbf a$ and $\mathbf b$.[3] Find $\overrightarrow{PQ}$ and show that it is a scalar multiple of $\overrightarrow{OP}$.[2] Hence prove that $O,P,Q$ are collinear, and state which point lies between the other two.[2]

Accepted Answer

(a) $\begin{pmatrix}-10\ -2\end{pmatrix}$, $\sqrt{104}\approx10.2$. (b) $\dfrac13\mathbf a+\dfrac23\mathbf b$. (c) $\dfrac12\overrightarrow{OP}$. (d) Collinear; P between O,Q.

Vector division and collinearity

Vector division and collinearity

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