Challenging +1.2 This is a multi-part Further Maths question requiring finding the line of intersection of two planes (via cross product of direction vectors), then finding a plane through this line and a point, and finally connecting to a system of equations. While it involves several steps and techniques (cross products, plane equations in different forms, interpreting solution sets), each individual step is fairly standard for FM students. The connection to infinite solutions at the end requires some insight but follows naturally from the geometry. Slightly above average difficulty due to length and FM content, but no particularly novel problem-solving required.
9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations
$$\begin{aligned}
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) ,
\end{aligned}$$
respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\).
Deduce, or prove otherwise, that the system of equations
$$\begin{aligned}
& 6 x - 5 y - 4 z = - 32 \\
& 5 x - y + 3 z = 24 \\
& 9 x - 2 y + 5 z = 40
\end{aligned}$$
has an infinite number of solutions.
9 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, which meet in the line $/$, have vector equations
$$\begin{aligned}
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) ,
\end{aligned}$$
respectively. Find a vector equation of the line $l$ in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.
Find a vector equation of the plane $\Pi _ { 3 }$ which contains $l$ and which passes through the point with position vector $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$. Find also the equation of $\Pi _ { 3 }$ in the form $a x + b y + c z = d$.
Deduce, or prove otherwise, that the system of equations
$$\begin{aligned}
& 6 x - 5 y - 4 z = - 32 \\
& 5 x - y + 3 z = 24 \\
& 9 x - 2 y + 5 z = 40
\end{aligned}$$
has an infinite number of solutions.
\hfill \mbox{\textit{CAIE FP1 2002 Q9 [12]}}