Moderate -0.3 This is a straightforward Further Maths vectors question requiring students to add two vectors to find AC, then apply the parallel condition (proportional components) to form two equations in p and q. The algebra is simple and the method is standard, making it slightly easier than average but still requiring proper technique.
Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by
$$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$
where \(p\) and \(q\) are constants.
Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\).
[3]
Vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are given by
$$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$
where $p$ and $q$ are constants.
Given that $\overrightarrow{AC}$ is parallel to $\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}$, find the value of $p$ and the value of $q$.
[3]
\hfill \mbox{\textit{SPS SPS FM 2020 Q1 [3]}}