Summary

Coordinate axes and planes. Space is referenced against three mutually perpendicular axes $OX,OY,OZ$ meeting at the origin $O$ (a right-handed system), giving three coordinate planes ($XY$, $YZ$, $ZX$) and eight octants; a point's coordinates $(x,y,z)$ are its signed perpendicular distances from these three planes.

Coordinates and distance.

derived by applying the Pythagorean theorem twice. Collinearity of three points is tested by checking the additive relation between their three pairwise distances.

Direction cosines and ratios. For a line making angles $\alpha,\beta,\gamma$ with the axes, direction cosines $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$ always satisfy

Direction ratios $(a,b,c)$ are any triple proportional to $(l,m,n)$; for the line joining $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$, direction ratios are $(x_2-x_1,\,y_2-y_1,\,z_2-z_1)$.

Equation of a line. Through a point $\vec a$ with direction $\vec b$:

Through two points, $\vec b$ (or $a,b,c$) is simply the coordinate/position difference between them.

Skew lines. Two lines are skew if they are neither parallel nor intersecting -- possible only in three (or more) dimensions, never in a plane. Tested algebraically: not parallel, and the system obtained by equating the two lines' general points is inconsistent.

Shortest distance.

$\text{Skew lines: } d=\left|\frac{(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)}{|\vec b_1\times\vec b_2|}\right| \qquad\qquad \text{Parallel lines: } d=\left|\frac{\vec b\times(\vec a_2-\vec a_1)}{|\vec b|}\right|$ …