Standard +0.3 This is a standard Further Maths vectors question requiring routine techniques: finding a plane equation via cross product of two vectors in the plane, finding line-plane intersection by substituting parametric equations, and calculating angle between line and plane using dot product formula. All steps are algorithmic with no novel insight required, making it slightly easier than average for FM content.
8 The points \(A , B , C\) have position vectors
$$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$
respectively, relative to the origin \(O\). Find a cartesian equation of the plane \(A B C\).
The point \(D\) has position vector \(6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\). Find the coordinates of \(E\), the point of intersection of the line \(O D\) with the plane \(A B C\).
Find the acute angle between the line \(E D\) and the plane \(A B C\).
8 The points $A , B , C$ have position vectors
$$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$
respectively, relative to the origin $O$. Find a cartesian equation of the plane $A B C$.
The point $D$ has position vector $6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$. Find the coordinates of $E$, the point of intersection of the line $O D$ with the plane $A B C$.
Find the acute angle between the line $E D$ and the plane $A B C$.
\hfill \mbox{\textit{CAIE FP1 2013 Q8 [11]}}