Challenging +1.3 This is a multi-part 3D vectors question requiring finding intersection points, using perpendicularity conditions, and calculating areas. While it involves several steps and techniques (solving simultaneous equations, dot products, distance formulas), each individual step follows standard A-level Further Maths procedures without requiring novel geometric insight. The AEA context and multi-part nature elevate it above average, but it's more computational than conceptually challenging.
The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } - 13 \\ 7 \\ - 1 \end{array} \right) + t \left( \begin{array} { c } 6 \\ - 2 \\ 3 \end{array} \right)\). The line \(L _ { 2 }\) passes through the point \(A\) with position vector \(\left( \begin{array} { c } 1 \\ p \\ 10 \end{array} \right)\) and is parallel to \(\left( \begin{array} { c } - 2 \\ 11 \\ - 5 \end{array} \right)\), where \(p\)
is a constant. The lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\).
Find
the value of \(p\),
the position vector of \(B\).
The point \(C\) lies on \(L _ { 1 }\) and angle \(A C B\) is \(90 ^ { \circ }\)
Find the position vector of \(C\).
The point \(D\) also lies on \(L _ { 1 }\) and triangle \(A B D\) is isosceles with \(A B = A D\).
\begin{enumerate}
\item The line $L _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { c } - 13 \\ 7 \\ - 1 \end{array} \right) + t \left( \begin{array} { c } 6 \\ - 2 \\ 3 \end{array} \right)$. The line $L _ { 2 }$ passes through the point $A$ with position vector $\left( \begin{array} { c } 1 \\ p \\ 10 \end{array} \right)$ and is parallel to $\left( \begin{array} { c } - 2 \\ 11 \\ - 5 \end{array} \right)$, where $p$\\
is a constant. The lines $L _ { 1 }$ and $L _ { 2 }$ intersect at the point $B$.\\
(a) Find\\
(i) the value of $p$,\\
(ii) the position vector of $B$.
\end{enumerate}
The point $C$ lies on $L _ { 1 }$ and angle $A C B$ is $90 ^ { \circ }$\\
(b) Find the position vector of $C$.
The point $D$ also lies on $L _ { 1 }$ and triangle $A B D$ is isosceles with $A B = A D$.\\
(c) Find the area of triangle $A B D$.\\
\hfill \mbox{\textit{Edexcel AEA 2017 Q3 [13]}}