Find the distance between tfrac x1=tfracy-11=tfrac z2 and…

Find the distance between $\tfrac x1=\tfrac{y-1}1=\tfrac z2$ and $\tfrac{x-1}1=\tfrac y1=\tfrac{z-1}2$.

Explanation of Method

The distance between two parallel lines.

Step 1: Solution

The common direction is $\vec p=\{1,1,2\}$. Take $M_1(0,1,0)$, $M_2(1,0,1)$, $\overrightarrow{M_1M_2}=\{1,-1,1\}$, which is not parallel to $\vec p$, so the lines are distinct:

\[ \vec p\times\overrightarrow{M_1M_2}=\{3,1,-2\},\qquad d=\frac{\sqrt{14}}{\sqrt6}=\sqrt{\frac73} . \]

\[d=\frac{\sqrt{21}}{3}\approx1.528\]

$M_1$ $M_2$ $d=\tfrac{\sqrt{21}}3$ $\ell_1$ $\ell_2$

Two parallel lines with direction $\{1,1,2\}$; the distance is the distance from $M_2$ to $\ell_1$.

Result

\[\boxed{\,d=\frac{\sqrt{21}}{3}\approx1.528\,}\]

Remark

The source's last check (“$\cos^2\alpha+\dots$ for the cross-product direction”) is meaningless and has been omitted.