Challenging +1.8 This is a multi-part Further Maths question requiring vector equations of lines, cross products to find perpendicular directions, the skew lines distance formula, and plane equations. While the techniques are standard for FM students, the question demands careful execution across multiple steps and integration of several vector concepts, placing it well above average difficulty.
9 With \(O\) as origin, the points \(A , B , C\) have position vectors
$$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$
respectively. Find a vector equation of the common perpendicular of the lines \(A B\) and \(O C\).
Show that the shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 2 } { 5 } \sqrt { } 5\).
Find, in the form \(a x + b y + c z = d\), an equation for the plane containing \(A B\) and the common perpendicular of the lines \(A B\) and \(O C\).
9 With $O$ as origin, the points $A , B , C$ have position vectors
$$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$
respectively. Find a vector equation of the common perpendicular of the lines $A B$ and $O C$.
Show that the shortest distance between the lines $A B$ and $O C$ is $\frac { 2 } { 5 } \sqrt { } 5$.
Find, in the form $a x + b y + c z = d$, an equation for the plane containing $A B$ and the common perpendicular of the lines $A B$ and $O C$.
\hfill \mbox{\textit{CAIE FP1 2006 Q9 [11]}}