Point to plane: d = |ax0+by0+cz0-d|/sqrt(a^{2+b}^{2+c}^2). This is the most frequently used distance formula.

Point to line: For line through A with direction b, distance from P is |AP x b|/|b|. The foot of perpendicular from P to the line is found by: parametric point Q = A + tb, then AP.b = 0 gives t = AP.b/|b|^2.

Between parallel planes: ax+by+cz=d1 and ax+by+cz=d2 have distance |d1-d2|/sqrt(a^{2+b}^{2+c}^2).

Between parallel lines: r=a1+tb and r=a2+sb with same direction b: d = |(a2-a1) x b|/|b|.

Between skew lines: d = |(a2-a1).(b1 x b2)|/|b1 x b2|. If b1 x b2 = 0 (parallel), use the parallel line formula instead.

Image of a point in a plane: Point P reflected across plane gives image P'. The midpoint of PP' lies on the plane, and PP' is perpendicular to the plane. Algorithm: (1) Write the line from P perpendicular to plane. (2) Find the foot F of perpendicular. (3) Image = 2F - P.

Distance of a point from a line measured along a specific direction: parametrize the given direction from the point, find where it hits the line (or a related plane), compute the distance.

Projection of a segment on a line: The length of the projection of vector v on direction d is |v.d|/|d|.