Chapter 2.5, Problem 7SP

Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes?

Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot

product.

The symmetric forms of two lines, $L_1$ and $L_2,$ are

$\begin{array}{l}{L}_1:\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}\\ L_2:\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}\end{array}$

You are to develop a formula for the distance $d$ between these two lines, in terms of the values

$a_1,b_1,c_1;b_2,c_2;x_1,y_1,z_1;$ and $x_2,y_2,z_2.$ The distance between two lines is usually taken to mean the

minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines.

7. Is your general expression valid when the lines are parallel? If not, why not? (Hint: What do you know about the value of the cross product of two parallel vectors? Where would that result show up in your expression for $d?$ )