2. Three forces, $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a rigid body. $\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$ N and $\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }$, where $p , q$ and $r$ are constants. All three forces act through the point with position vector ( $3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ ) m, relative to a fixed origin. The three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are equivalent to a single force ( $5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ ) N, acting at the origin, together with a couple $\mathbf { G }$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p , q$ and $r$.
\item Find $\mathbf { G }$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M5 2002 Q2 [6]}}