The position vectors of points $P$ and $R$ with respect to the origin $O$ are $\mathbf{p}$ a

Question

The position vectors of points $P$ and $R$ with respect to the origin $O$ are $\mathbf{p}$ and $\mathbf{r}$ respectively, and $\mathbf{q}$ is the position vector of the point $Q$ on $\overrightarrow{PR}$ such that $2\overrightarrow{PQ}=\overrightarrow{QR}$. Write down $\mathbf{q}$ in terms of $\mathbf{p}$ and $\mathbf{r}$.[1] It is given that $OR=3OP$ and the angle between $\mathbf{p}$ and $\mathbf{r}$ is $\theta $ where $\cos \theta =\frac{4}{5}$. $F$ is the foot of the perpendicular from $Q$ to the line $OP$. By using a suitable scalar product, find the exact value of $OF$ in terms of $\left| \mathbf{p} \right|$.[4] Hence determine if $F$ lies within the line segment $OP$, justifying your answer.[1]

Accepted Answer

(i) $\overrightarrow{OQ}=\frac{2}{3}\mathbf{p}+\frac{1}{3}\mathbf{r}$ (b)(i) $\frac{22}{15}\left| \mathbf{p} \right|$ (ii) Since $OF=\frac{22}{15}\left| \mathbf{p} \right|$, where $\frac{22}{15}>1$, therefore $F$ does not lie within the line segment $OP$.

2023 HCI Promo Q5

2023 HCI Promo Q5

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