Skew lines, what shortest distance can mean when two lines never meet, and computing it in both the skew and parallel cases

0:00In the plane, two lines either cross or run parallel. There is no third option, and that is such a reliable fact that almost nobody notices it is a fact about the plane rather than about lines. In space it is false. Two lines in space can cross. Two lines in space can run parallel. And two lines in space can do neither — pass each other at different heights, never meeting, never parallel.

0:32Each case answers the question how far apart are they differently. If they cross, the answer is nothing at all. If they run parallel, the answer is the length of a perpendicular dropped from any point of one onto the other, and any point will do. The third case is the one this topic is about, and it is the one with no answer you can write down without doing some work.

1:01The third case has a name. Two lines that neither meet nor run parallel are called skew. There is a second way to say the same thing, and it is the one worth carrying: skew lines do not lie in any one plane. Those two descriptions really are equivalent, and the argument is one sentence each way. If two lines lie in one plane, then inside that plane they are two ordinary lines, so they cross or run parallel. And if they cross, the two of them span a plane; if they run parallel, the two of them span a plane. So lying in a plane and being non-skew are the same condition.

1:45That is worth stating, because most treatments assert the equivalence and move on. It also explains the word skew is not doing any work. Skew does not mean slanted. It means neither of the two things you already know about. Counted on a grid of six thousand and eighty four pairs of lines: four hundred and sixty eight run parallel, one thousand two hundred and twenty four meet, and four thousand three hundred and ninety two do neither. Three cases, nothing left over.

2:20And a plane holds both lines at exactly the sixteen hundred and ninety two that are parallel or meeting — checked by solving the two lines' equations by hand, with no formula anywhere in it, and disagreeing at none. Skew pairs are not exotic. There is one in every room you have ever been in. Take a rectangular room, one unit deep, three units wide and two units high, with the origin at a floor corner.

2:53The first line runs diagonally across the ceiling: from the ceiling corner directly above the origin, to the ceiling corner diagonally opposite it. The second runs diagonally down a wall: from the ceiling corner above the far corner of the first axis, down to the floor corner diagonally across that wall. Read the directions straight off the corners. The ceiling diagonal goes one along, three across and nothing up, so its direction is one, three, nought. The wall diagonal goes nothing along, three across and two down, so its direction is nought, three, minus two.

3:37Those are the two lines. Now check both halves. First half: are they parallel? Parallel means the two triples stand in the same ratio. One over nothing is already impossible, and the third entries are nothing against minus two. The two triples are not proportional, so the lines are not parallel. Second half: do they meet? Write a point of each and set them equal. A point of the first is the parameter, three times the parameter, two. A point of the second is one, three times its own parameter, two minus twice that parameter.

4:20Take the third coordinates: two equals two minus twice the second parameter, so the second parameter is nothing. Take the second coordinates: three times the first parameter equals nothing, so the first parameter is nothing too. Now the first coordinates. The left side is the first parameter, which we have just shown is nothing. The right side is one. So nothing equals one, which it does not. There is no pair of parameters that works, so the lines never meet. Neither parallel nor meeting: skew, and both halves are checked rather than asserted.

5:02The answer, when we get to it, will be six sevenths. Now the question. How far apart are two lines that never meet? Be precise about what is being asked, because it is easy to be vague here. Take any point of the first line and any point of the second, and join them. That join has a length. Do it again with different points and you get a different length.

5:29There are infinitely many such joins, and the shortest distance is the smallest of all their lengths. That sentence hides an assumption, and it is worth saying out loud that we are taking it on trust: that among infinitely many lengths there is a smallest one, and that it is actually reached rather than merely approached. It is true here. But a student who wonders whether the smallest might never be attained has asked a good question, not a confused one, and it deserves to be named rather than glossed over.

6:06One more thing to be clear about. The two points that realise the minimum are generally not labelled in any question, are not the anchors of the two equations, and are harder to find than the answer itself. We will never need them. Here is the one thing about the shortest join that has a real argument behind it, and it takes fifteen seconds. Claim: the join that realises the minimum must be at right angles to both lines.

6:39Suppose it were not. Suppose the shortest join meets the second line at some angle that is not a right angle. Slide that endpoint a little along the second line, towards the foot of the perpendicular. The join gets shorter — because in any triangle the perpendicular is the shortest way from a point to a line. But we said the join was already the shortest. So it cannot have been at any angle other than a right angle.

7:10The same argument runs on the other end, along the first line. So the minimising segment is square to both. Notice that square to ONE line is not enough. Square to one line gives the shortest join from a fixed point to that line, which is a different and easier question. Both conditions together are what pins the segment down. That was checked exactly. Solving for the two parameters where the join is square to both, over the whole grid: the feet exist at every one of the five thousand six hundred and sixteen non-parallel pairs and at no parallel pair at all, and the join there is square to both directions at all five thousand six hundred and sixteen.

8:00And it really is the smallest: at every skew pair on the grid the join at the feet is strictly shorter than the join at any other pair of parameters tried — two hundred and thirteen thousand eight hundred and sixty four comparisons, shorter at every single one. Before the algebra, one honest word about the picture. Every drawing of a skew pair marks four points: two on one line, two on the other. And every such drawing puts those four points on a flat page.

8:37Four points, two on each of two skew lines, cannot lie in one plane. If they did, both lines would lie in that plane, and then the pair would not be skew at all. So the figure is not just simplified. It is showing you a configuration that cannot exist, and it is doing so unavoidably, because a page is flat. This matters more than it sounds. Students look at the flat quadrilateral, conclude that the two lines obviously lie in one plane, and then quietly decide the definition of skew is inconsistent.

9:17The fix is to turn the picture. Rotate the same four points and the depth appears: the two lines pass at different heights, and the quadrilateral was never flat. That is the one thing a moving picture can do here that a still one cannot, and it is worth the twenty seconds. Now build the thing. We know the shortest join is square to both directions. That is enough to find its direction without knowing either of its endpoints.

9:50Given two directions that are not parallel, there is a standard way to produce a third at right angles to both of them: cross them together. That result is square to both factors, and it is not the nought arrow precisely because the two directions are not parallel. Which is exactly what being skew guarantees. The construction is available in this case and in no other. That fact is borrowed. It belongs to vector algebra, not to anything about lines, and most treatments use it here in a subordinate clause without ever stating it as a result.

10:32So the crossed directions point along the common perpendicular. Divide by their own length and you have a unit arrow pointing the way the shortest join runs. For the room, the two directions cross to minus six, two, three, whose length is exactly seven. So the unit arrow is minus six, two, three, all over seven. Here is the move the whole topic turns on. We do not know the two feet. But we do know the direction the shortest join runs in. And a length along a known direction is a projection.

11:12So take any join at all — from any point of the first line to any point of the second — and project it onto that unit direction. The shadow it casts is the shortest distance. Why? Because the arbitrary join, the shortest segment, and the sideways parts all sit in a right-angled arrangement. Everything that is not along the common perpendicular is at right angles to it, and contributes nothing to the shadow.

11:43In particular, take the two anchors — the two points already written into the equations. The join between them will do as well as any other. So the distance is the anchor gap dotted into the unit arrow along the crossed directions. Written out: the anchor gap dotted into the crossed directions, over the length of the crossed directions, and the whole thing inside modulus bars because a projection can come out either sign and a distance cannot.

12:18That step deserves a second look, because it is the reason the formula is usable at all. The two points went in arbitrary and came out of the answer completely. Watch it happen. Slide the first point anywhere along the first line: it moves by some multiple of the first direction. That multiple gets dotted into the crossed directions — and the crossed directions are square to the first direction, so it contributes nothing.

12:49Slide the second point anywhere along the second line and the same thing happens for the same reason. So the numerator does not move at all. Checked over the whole grid: six combinations of slides at each of four thousand three hundred and ninety two skew pairs, and the numerator is unchanged at every one of them. And a control, so that is not just a routine ignoring its inputs: move an anchor by one step in a direction that is NOT along its own line, and the numerator does change — at three thousand and forty eight of those same pairs, the rest being the ones whose lines happened to run that way.

13:34This is the step a student who misses it pays for. If you believe the formula needs the two feet of the perpendicular, you cannot start, because the feet are not given. Any two points will do, and that is what the cancellation buys. So here is the result. The distance is the size of the anchor gap dotted into the crossed directions, over the length of the crossed directions.

14:04And here is a warning about how you may see it written. One displayed version in circulation prints the second bracket of its numerator as the two anchor position vectors CROSSED together, where their DIFFERENCE belongs. The multiplication cross has been set where a minus sign should be, three characters from a genuine cross in the same line. It cannot be right, and you do not need to take my word for it. That version disagrees with the derivation it is the conclusion of, with the worked example that follows it, and with the summary that restates it — all three of which carry the difference.

14:48And it is a different number. On a grid of four thousand three hundred and ninety two skew pairs the crossed version agrees with the correct one at only three hundred and four, and differs at four thousand and eighty eight. On the standard worked pair it gives the wrong answer outright. The anchors are one, one, nought and two, one, minus one. Their difference is one, nought, minus one, giving a numerator of ten. Crossed together they give minus one, one, minus one, and a numerator of three.

15:26If a line has ever gone onto a formula sheet in this topic, it is this one. Copy the difference. Look at the numerator on its own. Two vectors crossed, then dotted into a third. That is a single number, and it has a meaning. Its size is the volume of the box those three arrows span — the two directions and the gap between the anchors. Read it that way and the formula reads as a sentence. The volume of the box, divided by the area of its base, is the height. And the height is the distance.

16:05Checked at all four thousand three hundred and ninety two skew pairs: the length of the crossed directions times the distance is exactly the size of the numerator. One consequence falls out for free. That number is nothing exactly when the three arrows lie flat — when there is no box — and the three arrows lie flat exactly when the two lines lie in one plane. So the formula returns nothing at all precisely when the lines meet. Among the pairs that are not parallel it returns nought at exactly the one thousand two hundred and twenty four that meet, disagreeing at none.

16:49Which means the same computation classifies the pair and measures it. You do not need to test for intersection first. Run the formula, and a nought is the test. In coordinates the same result gets written differently, and it gets treated as a second formula to memorise. It is not. The numerator becomes a three-row determinant. The top row is the three coordinate gaps between the two anchors. The two rows below it are the two direction-ratio triples.

17:26That is the same number. Expand the determinant along its top row and you get the anchor gap dotted into the crossed directions, term for term. Checked at all six thousand and eighty four pairs of the grid. The denominator becomes a square root of three squared brackets. Those brackets are the three entries of the crossed directions, written out, so the root is its length. Checked at all six hundred and seventy six pairs of directions.

18:00One result, two dresses. Nothing new is happening. Keep the outer modulus bars, though. A determinant can perfectly well come out negative — on the grid it does so at two thousand one hundred and ninety six of the six thousand and eighty four pairs — and a distance cannot. And while we are here: swapping which line you call the first changes nothing. The crossed directions change sign, the anchor gap changes sign, so the numerator changes sign twice. The same distance at all four thousand three hundred and ninety two skew pairs. If you and a friend get answers of opposite sign, neither of you is wrong.

18:48Now the parallel case, which needs a formula of its own — and the reason is usually not given. The reason is mechanical, and it is a good one. If the two directions are parallel, crossing them gives the nought arrow. The denominator of the skew formula is the length of that arrow. So the formula does not give a poor answer for a parallel pair; it has nothing underneath it at all.

19:17Checked: the crossed directions vanish at every one of the four hundred and sixty eight parallel pairs on the grid, and the formula refuses all four hundred and sixty eight. There is a geometric reason underneath the mechanical one, and it is the better one to carry. For a skew pair there is exactly one segment realising the minimum, and its two endpoints are pinned. For a parallel pair every perpendicular between the two lines has the same length — there are infinitely many of them and they are all equally short.

19:55So the parallel question is a different question. It has no distinguished segment to find, and any point of one line will do as a starting point. Checked: slide the starting point to five different places along its own line and the distance is unchanged at all four hundred and sixty eight parallel pairs. And a control — for a skew pair that is false at every single one of the four thousand three hundred and ninety two.

20:29So start again for the parallel case, and it is three lines. Take the two anchors and the join between them. Take the common direction. That join makes some angle with the direction. Drop the perpendicular. You now have a right-angled triangle whose hypotenuse is the join and whose height is the distance you want. So the distance is the length of the join times the sine of that angle.

20:58And there is a product that hands you exactly a length times a length times a sine: cross the direction into the join. Its length is the direction's length times the join's length times the sine. So divide by the direction's length and the sine survives with one length attached. The distance is the length of the direction crossed into the anchor gap, over the length of the direction. Checked against the ordinary distance from a point to a line, measured with no cross product anywhere: the two agree at all four hundred and sixty eight parallel pairs. Sixty of them come out at nought, which is the two lines being the same line.

21:47Two worked pairs, end to end. First, a skew pair. Anchors at one, one, nought and two, one, minus one; directions two, minus one, one and three, minus five, two. The anchor gap is one, nought, minus one. The crossed directions are three, minus one, minus seven, whose squared length is fifty nine. The numerator is three plus nothing plus seven, which is ten. So the distance is ten over the root of fifty nine.

22:20Worth noticing what was not done: nobody checked whether those lines meet, and nobody found the feet. Second, a parallel pair. Both directions are two, three, six — the same triple, character for character, which is the check the question wants and which is easy to skip. Anchors at one, two, minus four and three, three, minus five, so the gap is two, one, minus one. Cross the direction into the gap and you get minus nine, fourteen, minus four, whose squared length is two hundred and ninety three.

22:59The direction's own length is seven exactly. So the distance is the root of two hundred and ninety three, over seven. And note the check has to be for proportionality, not for equality. If one direction is a multiple of the other, the pair is still parallel and the skew formula still divides by nothing. Four more, quickly, and one of them is worth waiting for. Anchors one, two, one and two, minus one, minus one, with directions one, minus one, one and two, one, two. The gap is one, minus three, minus two; the crossed directions are minus three, nothing, three; the numerator is minus nine; and the distance is three over root two.

23:51Next, a pair in coordinate form, so the triples come off the denominators: anchors minus one, minus one, minus one and three, five, seven, with ratios seven, minus six, one and one, minus two, one. This one has a coincidence in it. The gap is four, six, eight, and the crossed directions come out at minus four, minus six, minus eight — exactly minus one times the gap. So the numerator is minus the squared length of the gap, and the whole quotient collapses to the length of the gap itself: two root twenty nine.

24:34Third: anchors one, two, three and four, five, six, directions one, minus three, two and two, three, one. The gap is three, three, three; the crossed directions are minus nine, three, nine; the numerator is nine; the distance is three over root nineteen. The fourth is the awkward one, and the difficulty is not the formula. Both lines are written with the parameter multiplied through the components, so the anchors and directions have to be unpicked before anything else can start.

25:12Once unpicked it is one, minus two, three along minus one, one, minus two, against one, minus one, minus one along one, two, minus two. The gap is nothing, one, minus four; the crossed directions are two, minus four, minus three; the numerator is eight; the distance is eight over root twenty nine. And the one worth waiting for. Anchors six, two, two and minus four, nothing, minus one; directions one, minus two, two and three, minus two, minus two.

25:49The gap is minus ten, minus two, minus three. The crossed directions are eight, eight, four, whose length is exactly twelve. The numerator is minus one hundred and eight. And one hundred and eight over twelve is nine — a whole number, the only one in the set. One habit to finish with, because it saves the most time. Classify before you compute. Three branches, and two of them need no formula at all.

26:21Are the two direction triples proportional? If yes, the lines are parallel, and you want the sine formula: the direction crossed into the anchor gap, over the length of the direction. If no, they are not parallel, and there are two cases left — but you do not have to separate them by hand. Run the skew formula. If it returns nought, the lines meet and the answer was nought. If it returns anything else, the lines are skew and that is your answer.

26:55So the decision really only has two branches that need work, and the third resolves itself. One caution on reading the branch. Proportional, not equal. Two directions can be parallel without sharing a single entry, and a pair like minus two, minus four, minus four against two, four, four is perfectly parallel. And one caution on reading the answer. The numerator can come out negative — most of the worked items above have a negative numerator — and the sign is discarded at the very end. Do not go hunting for an arithmetic slip that is not there.

27:40So, what is worth carrying out of this. Space has three cases where the plane has two. Meeting, parallel, and skew — and skew is exactly the case no single plane can hold. The shortest join between two skew lines is square to both of them, and that is not a definition, it is a one-line argument: a join that is not square to a line can be slid along that line and made shorter.

28:11Its direction is therefore the two directions crossed, which exists precisely because the pair is not parallel. And the formula is a projection. Any join at all, projected onto that direction — which is why the two anchors given in the question may simply be used, and why the two feet of the perpendicular never have to be found. The determinant version is that same number in coordinates, not a second result. Keep the outer bars: the determinant can come out negative and a distance cannot.

28:49The parallel case needs its own formula because crossing two parallel directions gives nothing to divide by — and because a parallel pair has infinitely many equal perpendiculars rather than one distinguished segment. A distance of nothing means the lines meet. And two skew lines do have an angle between them, by the same slide-them-to-one-point construction as any other pair — a fact that tends to appear only in a summary, and never where you would look for it.