Shortest Distance Between Two Straight Lines

Definition 6.6. Two lines are coplanar if they lie in a common plane. Two lines that are either parallel or intersecting are automatically coplanar.

Definition 6.7 (Skew lines). Two lines in space are skew if they are neither parallel nor intersecting — equivalently (by the Note above), skew lines are precisely the non-coplanar ones. Every pair of lines in space is either parallel or skew, once intersection is ruled out... more precisely, any two given lines are: intersecting, parallel-and-distinct, or skew.

If two lines intersect, the distance between them is $0$. If they are parallel (and distinct), the distance is the length of the common perpendicular segment; the shortest distance between two skew lines is likewise defined as the length of the line segment perpendicular to BOTH lines.

Theorem 6.13 (parallel lines). The shortest distance between the parallel lines $\vec r=\vec a+s\vec b$ and $\vec r=\vec c+t\vec b$ (same direction $\vec b\ne\vec 0$) is

(Idea: $d$ is the perpendicular leg of the right triangle formed by $\vec{AB}=\vec c-\vec a$ and the direction $\vec b$, so $d=|\vec c-\vec a|\sin\theta$, and $|(\vec c-\vec a)\times\vec b|=|\vec c-\vec a||\vec b|\sin\theta$ gives the stated formula after dividing by $|\vec b|$.)

Theorem 6.14 (skew lines). The shortest distance between the skew lines $\vec r=\vec a+s\vec b$ and $\vec r=\vec c+t\vec d$ is

(Idea: $\vec b\times\vec d$ is perpendicular to BOTH lines, so $\dfrac{\vec b\times\vec d}{|\vec b\times\vec d|}$ is the unit vector along the common perpendicular; $\delta$ is then the absolute value of the projection of $\vec{AC}=\vec c-\vec a$ onto that unit vector.) …