Points on two skew lines closest to one another

Question

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between which is minimum? That distance is thus but what are the points where it is achieved?

Answer 1

HINT: You're looking for points $P$ and $Q$ on the respective lines for which the vector $\overrightarrow{PQ}$ will be orthogonal (perpendicular) to both lines. (To understand why, think about the hypotenuse of a right triangle.) So, for starters, you want a vector orthogonal to both $\vec x_2-\vec x_1$ and $\vec x_4-\vec x_3$.