The projection of a in the plane normal to b:

This is this is a double cross:

The distance of a point p to a plane specified by 3 points a, b, c:

Form vectors p - a,  b - a,  c - a ;
Take their mixed doubles product and divide by the magnitude of the cross of  b - a and c - a, and the magnitude of the result is the answer.

Finding volumes of parallelopipeds:

This is the magnitude of the mixed cross of the three vectors defining its edges at a point.

Finding the minimum distance between two lines:

Find p, q on one and r, s on the other, take the magnitude of the mixed double product of p - r, p - q, r - s divided by the magnitude of the cross product of p - q and r - s.