Distance between two lines

Find the distance between two lines

l_1\Rightarrow r_1=\begin{pmatrix}1\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\1\\1\end{pmatrix}
l_2\Rightarrow r_2=\begin{pmatrix}-1\\3\\-1\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\-1\end{pmatrix}
Let A equal the point at which AB intersects with l_1
Let B equal the point at which AB intersects with l_2
\vec{OA}=\begin{pmatrix}1\\\lambda\\\lambda\end{pmatrix}
\vec{OB}=\begin{pmatrix}-1+2\mu\\3-\mu\end{pmatrix}
\vec{AB}=\vec{OB}-\vec{OA}=\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-\lambda\end{pmatrix}
d_1=\begin{pmatrix}0\\1\\1\end{pmatrix}
d_2=\begin{pmatrix}2\\-1\\-1\end{pmatrix}
\vec{AB}\cdot d_1=0
\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-lambda\end{pmatrix}\cdot\begin{pmatrix}0\\1\\1\end{pmatrix}=0
\vec{AB}\cdot d_2=0
\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-lambda\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\-1\end{pmatrix}=0

Question	Answer
What are the two direction vectors d_1 and d_2 for the lines l_1: \begin{pmatrix}1\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\1\\1\end{pmatrix} and l_2: \begin{pmatrix}-1\\3\\-1\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\-1\end{pmatrix}?	d_1=\begin{pmatrix}0\\1\\1\end{pmatrix} and d_2=\begin{pmatrix}2\\-1\\-1\end{pmatrix}
How is \vec{OA} expressed for the point on l_1?	\vec{OA}=\begin{pmatrix}1\\\lambda\\\lambda\end{pmatrix}
How is \vec{OB} expressed for the point on l_2?	\vec{OB}=\begin{pmatrix}-1+2\mu\\3-\mu\\-1-\mu\end{pmatrix}
How is \vec{AB} calculated from positions A and B?	\vec{AB}=\vec{OB}-\vec{OA}=\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-\lambda\end{pmatrix}
What condition is applied to \vec{AB} and direction vectors d_1, d_2 to find the shortest distance between l_1 and l_2?	\vec{AB}\cdot d_1=0 and \vec{AB}\cdot d_2=0
Write the dot product equation \vec{AB}\cdot d_1=0 using the vectors from the document.	\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-\lambda\end{pmatrix}\cdot\begin{pmatrix}0\\1\\1\end{pmatrix}=0
Write the dot product equation \vec{AB}\cdot d_2=0 using the vectors from the document.	\begin{pmatrix}-2+2\mu\\3-\mu-\lambda\\-1-\mu-\lambda\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\-1\end{pmatrix}=0