Challenging +1.8 This is a multi-part Further Maths question requiring systematic application of vector techniques: finding the common perpendicular to skew lines (involving cross products and simultaneous equations), determining a plane equation in vector form, and finding the intersection of two planes. While each component uses standard methods, the question demands sustained accuracy across multiple steps and comfortable manipulation of 3D vector geometry, placing it well above average difficulty.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations
$$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$
respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Show that the position vector of \(P\) is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and find the position vector of \(Q\).
Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), an equation of the plane \(\Pi\) which passes through \(P\) and is perpendicular to \(l _ { 1 }\).
The plane \(\Pi\) meets the plane \(\mathbf { r } = p \mathbf { i } + q \mathbf { j }\) in the line \(l _ { 3 }\). Find a vector equation of \(l _ { 3 }\).
The lines $l _ { 1 }$ and $l _ { 2 }$ have equations
$$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$
respectively. The point $P$ on $l _ { 1 }$ and the point $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. Show that the position vector of $P$ is $3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and find the position vector of $Q$.
Find, in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$, an equation of the plane $\Pi$ which passes through $P$ and is perpendicular to $l _ { 1 }$.
The plane $\Pi$ meets the plane $\mathbf { r } = p \mathbf { i } + q \mathbf { j }$ in the line $l _ { 3 }$. Find a vector equation of $l _ { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2016 Q11 EITHER}}