Question 6 - A-Level Maths

The points \(A , B\) and \(C\) have coordinates \(( 3,1 , - 6 ) , ( 5 , - 2,0 )\) and \(( 8 , - 4 , - 6 )\) respectively.
1. Show that the vector \(\overrightarrow { A C }\) is given by \(\overrightarrow { A C } = n \left[ \begin{array} { r } 1
  - 1
  0 \end{array} \right]\), where \(n\) is an integer.
2. Show that the acute angle \(A C B\) is given by \(\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)\).
Find a vector equation of the line \(A C\).
The point \(D\) has coordinates \(( 6 , - 1 , p )\). It is given that the lines \(A C\) and \(B D\) intersect.
1. Find the value of \(p\).
2. Show that \(A B C D\) is a rhombus, and state the length of each of its sides.