Question

has anyone ever looked at Problem Set 3, ex 1E-7

Answer 1

are you asking about
"Formulate a general method for finding the distance between two skew lines" ?

skew lines do not intersect, but are not parallel.
In general, they will have the equation in 3-Dim:
\[ p_1 + t\ \vec{v_1} \]
where \( p_0 \) is a point on the line, \( \vec{v_1} \) is a "direction vector" and t is a scalar that we can vary to "move" along the line.

the hint states:
the shortest line segment joining the two skew lines will be perpendicular to both.

use the cross product of the two lines direction vectors to find a vector perpendicular to both lines. Normalize this vector by its length to make it a unit length vector.

create a vector with its tail on one of the lines and its head on the other line.
if we dot product that vector with the unit length perpendicular vector, we will get the shortest distance between the 2 skew lines.