Given the vector equations for two lines $$(x_0,y_0,z_0) = (a,b,c)+t(d,e,f)$$ and $$(x_1,y_1,z_1) = (g, h, i) + t(j, k, l),$$
why is the shortest distance between the two lines not equal to $$\dfrac{\det\left(\begin{bmatrix} g - a & h - b & i - c \\ d & e & f \\ j & k & l \end{bmatrix}\right)}{||(d,e,f)\times(j,k,l)||}?$$
My thinking was that this would be the same as taking a parallelopiped's volume and dividing by the area of the base, thus giving the height (the distance between the two lines). However, I tried this out with a couple examples and my answers differed from the actual results.
You formula is correct. What is the counter example?