3.
\section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{1fa7336c-20aa-45c1-b685-d8e205367227-3_528_755_317_618}
Figure 1 shows a box in the shape of a cuboid \(P Q R S T U V W\) where \(\overrightarrow { P Q } = 3 \mathbf { i }\) metres, \(\overrightarrow { P S } = 4 \mathbf { j }\) metres and \(\overrightarrow { P T } = 3 \mathbf { k }\) metres. A force \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts at \(Q\), a force \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at \(R\), a force \(( - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(T\), and a force \(( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(W\). Given that these are the only forces acting on the box, find
- the resultant force acting on the box,
- the resultant vector moment about \(P\) of the four forces acting on the box.
When an additional force \(\mathbf { F }\) acts on the box at a point \(X\) on the edge \(P S\), the box is in equilibrium.
- Find \(\mathbf { F }\).
- Find the length of \(P X\).