# Points on two skew lines closest to one another

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?

• The title is misleading or confusing, as skew lines do not intersect :) You're asking about the distance between the skew lines, or the points on the respective skew lines that are closest to one another. – Ted Shifrin Jun 29 '15 at 22:01
• @TedShifrin I meant the points on the respective skew lines that are closest to one another. – user_1_1_1 Jun 29 '15 at 22:03
• Yup, much better title; thanks. See if my hint helps. – Ted Shifrin Jun 29 '15 at 22:04
• – Ujjwal Rajput Mar 18 '16 at 10:45
• – Ujjwal Rajput Mar 18 '16 at 10:53

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 $\vec x_4-\vec x_3$.
• Yup that hint completely solves my problem thanks, but i will wait to see if someone else actually solves and answers $P,Q$ in terms of $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$? – user_1_1_1 Jun 29 '15 at 22:19
• Well, work it out! Parametrize the lines by $\vec\alpha(t)=\vec x_1+t(\vec x_2-\vec x_1)$ and $\vec\beta(s)$ analogous;y, and solve for the values of $s,t$ for which $\vec\alpha(t)-\vec\beta(s)$ is orthogonal to both $\vec x_2-\vec x_1$ and $\vec x_4-\vec x_3$. With numbers this won't be too bad. With all these letters, a bit less fun. – Ted Shifrin Jun 29 '15 at 22:22