\includegraphics{figure_6}

The diagram shows a solid figure $ABCDEF$ in which the horizontal base $ABC$ is a triangle right-angled at $A$. The lengths of $AB$ and $AC$ are 8 units and 4 units respectively and $M$ is the mid-point of $AB$. The point $D$ is 7 units vertically above $A$. Triangle $DEF$ lies in a horizontal plane with $DE$, $DF$ and $FE$ parallel to $AB$, $AC$ and $CB$ respectively and $N$ is the mid-point of $FE$. The lengths of $DE$ and $DF$ are 4 units and 2 units respectively. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $\overrightarrow{AB}$, $\overrightarrow{AC}$ and $\overrightarrow{AD}$ respectively.

\begin{enumerate}[label=(\roman*)]
\item Find $\overrightarrow{MF}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$. [1]

\item Find $\overrightarrow{FN}$ in terms of $\mathbf{i}$ and $\mathbf{j}$. [1]

\item Find $\overrightarrow{MN}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$. [1]

\item Use a scalar product to find angle $FMN$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2019 Q6 [7]}}