Since \(R_1\) is a point on \(\vec\ell_1\text{,}\) we know that
\begin{equation}
R_1 = (1+3t, 2-t, t), \quad \text{ for some real number } t\text{,}\tag{2.5.4}
\end{equation}
and similarly,
\begin{equation}
R_2 = (-2+4s, 3+s, 5+2s), \quad \text{ for some real number } s\text{.}\tag{2.5.5}
\end{equation}
The vector \(\overrightarrow{R_1R_2}\) is therefore given by
\begin{equation*}
\overrightarrow{R_1R_2}=\bbm -3+4s-3t\\ 1+s+t\\ 5+2s-t\ebm\text{,}
\end{equation*}
for some pair of real numbers \(s\) and \(t\text{.}\)
We know that the line segment \(\overline{R_1R_2}\) must be perpendicular to both \(\vec\ell_1\) and \(\vec\ell_2\) in order to minimize the distance, so the vector \(\overrightarrow{R_1R_2}\) must be orthogonal to both \(\vec{d}_1\) and \(\vec{d}_2\text{.}\) Thus,
\begin{align*}
0 = \vec{d}_1\boldsymbol{\cdot}\overrightarrow{R_1R_2} \amp = 3(-3+4s-3t)-1(1+s+t)+1(5+2s-t) \\
\amp =13s-11t-5, \text{ and}\\
0 = \vec{d}_2\boldsymbol{\cdot}\overrightarrow{R_1R_2} \amp = 4(-3+4s-3t)+1(1+s+t)+2(5+2s-t)\\
\amp = 21s-13t-1\text{.}
\end{align*}
We end up having to solve a system of two linear equations in the two variables, \(s\) and \(t\text{,}\) given by
\begin{equation*}
\begin{array}{ccccc}
13s \amp - \amp 11t \amp = \amp 5\\
21s \amp - \amp 13t \amp = \amp 1
\end{array}\text{.}
\end{equation*}
You probably had to solve such systems in high school. One option is to solve graphically, by plotting the lines given by each equation, and seeing where they intersect. However, this method has little hope of providing an accurate answer. Instead, we try a little algebra.
Multiplying the first equation by 21 and the second by 13 gives us the equations
\(273s-231t=105\) and
\(273s-169t=13\text{,}\) respectively. Subtracting the second equation from the first, we have
\(-62t = 92\text{,}\) so
\(t=-\frac{92}{62}=-\frac{46}{31}\text{.}\) Plugging this value back into any of the previous equations gives us
\(s=-\frac{351}{403}=-\frac{27}{31}\text{.}\) (We didnβt promise that the numbers would work out nicely!) Plugging these values back into equations
(2.5.4) and
(2.5.5), we find
\begin{equation*}
R_1 = \left(-\frac{107}{31}, \frac{108}{31}, -\frac{46}{31}\right) \quad \text{ and } \quad R_2 = \left(-\frac{170}{31}, \frac{66}{31}, \frac{101}{31}\right)\text{.}
\end{equation*}
Our vector \(\overrightarrow{R_1R_2}\) is then given by
\begin{equation*}
\overrightarrow{R_1R_2} = \bbm -\frac{63}{31}\\ -\frac{42}{31}\\ \frac{147}{31}\ebm = \frac{1}{31}\bbm -63\\ -42\\ 147\ebm\text{,}
\end{equation*}
and the distance between the two lines is given by
\begin{equation*}
\norm{\overrightarrow{R_1R_2}} = \frac{1}{31}\sqrt{63^2+42^2+147^2} = \frac{42}{\sqrt{62}},
\end{equation*}
as before.