- With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.
The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\)
The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).
- Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5
The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226 - find the possible values of the constant \(k\).