Do the lines tfrac x1=tfrac y2=(z-1)/(1) and tfrac…

Do the lines $\tfrac x1=\tfrac y2=\tfrac{z-1}{1}$ and $\tfrac x2=\tfrac{y-1}1=\tfrac z2$ intersect?

Explanation of Method

Deciding whether two lines intersect: here they are skew.

Here $M_1(0,0,1)$, $\vec d_1=\{1,2,1\}$, $M_2(0,1,0)$, $\vec d_2=\{2,1,2\}$: \[\begin{vmatrix}0&-1&1\\1&2&1\\2&1&2\end{vmatrix}=0+0-3=-3\ne0 .\]

\[\text{no intersection: the lines are skew}\]

Parametric check: $s=2t$ and $1+s=2t$ would give $1=0$.

\[ d=\frac{|[\overrightarrow{M_1M_2},\vec d_1,\vec d_2]|}{|\vec d_1\times\vec d_2|}=\frac{3}{|\{3,0,-3\}|}=\frac{3}{3\sqrt2}=\frac{\sqrt2}{2} . \]

$d=\tfrac1{\sqrt2}$ $\ell_1$ $\ell_2$

The skew lines $\ell_1,\ell_2$ and their common perpendicular (dashed), of length $\tfrac{1}{\sqrt2}$.

\[\boxed{\,\text{no intersection: the lines are skew}\,}\]