The line \(L_1\) has vector equation
$\displaystyle
{\bf r} =
\begin{pmatrix}
1 \\
0 \\
2
\end{pmatrix}
+
\lambda
\begin{pmatrix}
\hphantom{-} 2 \\
\hphantom{-} 2 \\
-3
\end{pmatrix}
$.
The line \(L_2\) has vector equation
$\displaystyle
{\bf r} =
\begin{pmatrix}
\hphantom{-} 4 \\
-2 \\
\hphantom{-} 9
\end{pmatrix}
+
\mu
\begin{pmatrix}
\hphantom{-} 1 \\
\hphantom{-} 2 \\
-2
\end{pmatrix}
.
$
Show that the distance \(D\)
between a point on \(L_1\) and a point on \(L_2\)
can be expressed in the form
\[
D^2 = \left(3\mu -4 \lambda-5 \right)^2 + \left( \lambda -1 \right)^2 + 36\,.
\]
Hence determine the minimum distance
between these two lines and find the coordinates
of the points on the two lines that are the minimum distance apart.
The line \(L_3\) has vector equation
${\bf r} =
\begin{pmatrix}
2 \\
3 \\
5
\end{pmatrix}
+
\alpha
\begin{pmatrix}
0 \\
1 \\
0
\end{pmatrix}
.
$
The line \(L_4\) has vector equation
$
{\bf r} =
\begin{pmatrix}
\hphantom{-} 3 \\
\hphantom{-} 3 \\
-2
\end{pmatrix}
+
\beta
\begin{pmatrix}
\, 4k\\
1-k \\
\!\!\! -3k
\end{pmatrix}
.
$
Determine the minimum distance between these two lines,
explaining geometrically the two different cases that arise
according to the value of \(k\).