Challenging +1.2 This is a structured multi-part question on vectors in 3D geometry involving a cuboid. While it requires understanding of cuboid properties, vector operations, and likely the scalar triple product for volume, the geometric setup is well-defined and the question guides students through the problem. It's moderately harder than average due to the 3D visualization required and multiple techniques needed, but the cuboid context provides strong scaffolding compared to abstract vector problems.
3.The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid.With respect to a fixed origin \(O\) ,the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and \(\mathbf{d}\) respectively,where
$$\mathbf { a } = \left( \begin{array} { c }
1 \\
2 \\
- 1
\end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c }
0 \\
- 3 \\
- 8
\end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c }
4 \\
- 1 \\
- 10
\end{array} \right)$$
3.The points $A , B , C , D$ and $E$ are five of the vertices of a rectangular cuboid and $A E$ is a diagonal of the cuboid.With respect to a fixed origin $O$ ,the position vectors of $A , B , C$ and $D$ are $\mathbf { a , b , c }$ and $\mathbf{d}$ respectively,where
$$\mathbf { a } = \left( \begin{array} { c }
1 \\
2 \\
- 1
\end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c }
0 \\
- 3 \\
- 8
\end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c }
4 \\
- 1 \\
- 10
\end{array} \right)$$
\hfill \mbox{\textit{Edexcel AEA 2016 Q3 [9]}}