Distance between parallel traces of a plane on the coordinate planes

Problem

A plane in space meets $x=0$ in the line $L_1$ and $z=0$ in the line $L_2$. It is given that $L_1\parallel L_2$, that $L_1$ passes through $(0,2,-11)$, and that $L_2$ passes through $(8,21,0)$. Find the distance between the two lines, in simplest radical form.

The full question is on the CEEC 114 AST Mathematics I paper.

Where it goes wrong

It is easy to treat the two traces as skew lines and force a formula, or to start from a general plane equation and slip on the parallel condition. The key is simpler. One line lies in $x=0$, the other in $z=0$, and they must be parallel, so there is almost no choice left for the direction vector.

Idea

A direction vector of $L_1\subset \{x=0\}$ satisfies $u_x=0$. A direction vector of $L_2\subset \{z=0\}$ satisfies $u_z=0$. The lines are parallel, so they share the same $\mathbf{u}$, and therefore

The distance between two parallel lines is the length of the component of the segment joining the two known points that is perpendicular to $\mathbf{u}$. Removing the $y$ direction leaves

(The fill-in digits are $1$, $8$, $5$.)

What the figure shows

• The translucent blue and purple planes are $x=0$ and $z=0$. Their traces are $L_1$ and $L_2$.
• The gold line is the common perpendicular segment. Its length stays at $\sqrt{185}$.
• Drag the canvas or use the view sliders. The picture changes, but the parallel lines and the distance do not.