Find the 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