Distance between two skew lines

Question

I have 2 skew lines $L_A$ and $L_B$ and 2 parallel planes $H_A$ and $H_B$.

The line $L_A$ lies in $H_A$ and $L_B$ in $H_B$. If the equations of $H_A$ and $H_B$ are given like this:

$x+y+z = 0$ (for $H_A$)

$x+y+z = 5$ (for $H_B$)

Can I just simply say that the distance between two lines $L_A$ and $L_B$ is 5 since there the two planes they lies are separated apart by 5?

Answer 1

No. The distance between the two planes is not 5 in the first place. However, if you find the correct distance between the two planes, then your answer may still be wrong if the lines are parallel. If they are not parallel, then it happens to be correct. You gave the condition of skew lines, but I mention these two cases because it shows that it is not at all trivial why the distance should be as claimed, and there is something crucial about the lines being skew.

Answer 2

Yes,the shortest distance between the skew lines will equal the distance between the planes.