[conservatism_IS_compassion]

Given the points A, B, C, and D in three-space.

What is the closest approach of the line passing through points A and B to the line passing through the points C and D?

Took me a couple of hours to see how to use dot products and cross products to solve it (and I'm a guy). As always: "This problem, when solved, will be simple."

[conservatism_IS_compassion]

What is the closest approach of the line passing through points A and B to the line passing through the points C and D?

"This problem, when solved, will be simple."

Only when I visualized the lines from the POV looking perpendicular to the line connecting the points of closet approach could I solve the problem.

Then I saw that the direction of that line would be defined by the cross product of the vectors AB and CD. Call the resulting vector V. But since we only want the direction of that vector and not its magnitude, we calculate the square root of the dot product of vector V with itself, and divide each component of V by the resulting magnitude of V. Call the result vector W.

Looking at the vector W from a direction perpendicular to it, we see that the dot product of vector W with any vector which connects line AB with line CD (e.g., AC or BD or AD or BC) yields the shortest distance between lines AB and CD.