Moderate -0.3 This is a straightforward vectors question testing basic concepts: collinearity via scalar multiples, ratio calculations, finding points on a line with given distance from origin, and perpendicular vectors. All parts use standard techniques with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-part nature and need for careful geometric reasoning in parts (b) and (c).
The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\), \(\begin{pmatrix} -3 \\ 6 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 12 \end{pmatrix}\) respectively.
Show that \(B\) lies on \(AC\). [2]
Find the ratio \(AB : BC\). [1]
The diagram shows the line \(x + y = 6\) and the point \(P(2, 4)\) that lies on the line.
A copy of the diagram is given in the Printed Answer Booklet.
\includegraphics{figure_1}
The distinct point \(Q\) also lies on the line \(x + y = 6\) and is such that \(|\overrightarrow{OQ}| = |\overrightarrow{OP}|\), where \(O\) is the origin.
Find the magnitude and direction of the vector \(\overrightarrow{PQ}\). [3]
The point \(R\) is such that \(\overrightarrow{PR}\) is perpendicular to \(\overrightarrow{OP}\) and \(|\overrightarrow{PR}| = \frac{1}{2}|\overrightarrow{OP}|\).
Write down the position vectors of the two possible positions of the point \(R\). [2]
\begin{enumerate}[label=(\alph*)]
\item The points $A$, $B$ and $C$ have position vectors $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$, $\begin{pmatrix} -3 \\ 6 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 12 \end{pmatrix}$ respectively.
\begin{enumerate}[label=(\roman*)]
\item Show that $B$ lies on $AC$. [2]
\item Find the ratio $AB : BC$. [1]
\end{enumerate}
\item The diagram shows the line $x + y = 6$ and the point $P(2, 4)$ that lies on the line.
A copy of the diagram is given in the Printed Answer Booklet.
\includegraphics{figure_1}
The distinct point $Q$ also lies on the line $x + y = 6$ and is such that $|\overrightarrow{OQ}| = |\overrightarrow{OP}|$, where $O$ is the origin.
Find the magnitude and direction of the vector $\overrightarrow{PQ}$. [3]
\item The point $R$ is such that $\overrightarrow{PR}$ is perpendicular to $\overrightarrow{OP}$ and $|\overrightarrow{PR}| = \frac{1}{2}|\overrightarrow{OP}|$.
Write down the position vectors of the two possible positions of the point $R$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q2 [8]}}