Direction ratios as any proportional triple, and normalising back to cosines

0:00Direction cosines pin a direction down perfectly. So why would anyone want anything else? Because of what they look like. Here is a perfectly ordinary direction, written in cosines: minus nine over eleven, six over eleven, minus two over eleven. And here is the same direction written another way: minus eighteen, twelve, minus four. Three whole numbers. No fractions, no roots, nothing to carry around. And every piece of information in the first triple is still in the second one.

0:39That second kind of triple is what this video is about. They are called direction ratios, and the reason to have them is exactly the reason you just saw: a triple that arrives from somewhere — from two points, from an equation, from a picture — arrives unscaled, and there is no reason to tidy it up until you actually want an angle. So, the definition, and it is short.

1:08Three numbers a, b and c are direction ratios of a line when they are proportional to its direction cosines. Proportional means there is one number — call it k — that turns the cosines into the ratios. A equals k times l. B equals k times m. C equals k times n. Now look hard at that k. It is the same k in all three lines. One multiplier, doing all three at once.

1:41That is not a technicality, it is the entire definition, and it is the place people go wrong. If you double the first entry to make it tidier, and triple the second because that looked tidier too, you have not simplified anything. You have written down a triple for a completely different line. One multiplier. All three entries. Every time. Here is the consequence, and it is the thing that makes ratios a different kind of object from cosines.

2:16K can be any non-zero number you like. Positive, negative, whole, fractional. Every choice gives another perfectly good ratio triple for the same line. So one line has endlessly many direction ratio triples. Compare that with cosines. One line has exactly two cosine triples — one for each way along it — and that is the whole supply. Endlessly many of one, exactly two of the other. They are not the same kind of thing at all, and calling ratios untidy cosines gets this exactly backwards.

2:57But there is something all of those endlessly many triples agree on. Take any of them, divide by its own length, and you land on one of the same two cosine triples every time. The common factor cancels, top and bottom, and vanishes. That invariance is what makes the whole family usable. One quick note before we go on, because it saves confusion later. You will sometimes see these called direction numbers instead of direction ratios.

3:32Same triple. Same object. Different name. If you meet the other wording somewhere, nothing new is being introduced. Now the piece of work that matters: getting the cosines back out of a ratio triple. The answer is a one-line formula and you have probably seen it. But it is worth deriving, because the derivation is the only place the identity from the last topic ever gets used, and the derivation is where the interesting part hides.

4:07Start from the definition, written the other way up. L over a equals m over b equals n over c, and all three equal some common constant. Call it k. So l is k times a. M is k times b. N is k times c. Now square all three and add them. The left-hand side is l squared plus m squared plus n squared — and that is one. That is the identity, and it is the only fact we need from it.

4:41The right-hand side is k squared times the quantity a squared plus b squared plus c squared. So k squared times that sum is one. Which means k squared is one over the sum. Which means k is plus or minus one over the square root of the sum. Substitute back and there is the answer: l is a over the root of a squared plus b squared plus c squared, and the same for the other two.

5:11Notice what that root is. It is the length of the triple you were handed. Nothing more mysterious than that. Now, the plus or minus, which is the part that gets dropped. The square root gave two values for k, and both of them are legitimate. So the recovery has two answers, not one, and they differ by an overall sign. This is not a loose end in the algebra. It is the geometry showing up in the arithmetic.

5:45A ratio triple carries a line. It does not carry a sense of travel along that line — remember, k could be negative, and flipping k's sign flips every entry. A line has two senses. So recovering cosines from ratios has to give you two triples: one for each arrowhead. And they are negatives of each other, exactly as they were in the last topic. The version most people end up memorising has no sign in front of it. If you learn it that way you will produce one of the two answers and never know the other one existed.

6:26And the sign is not decoration. If a question tells you the line makes an obtuse angle with some axis, or fixes a direction of travel, that information picks the sign — and you cannot use it if you have thrown the choice away. Two worked ones. Both come out clean, which is not an accident — they were chosen that way. First: ratios two, minus one, minus two. Square and add: four plus one plus four is nine. The root of nine is three, exactly.

7:02So the cosines are two thirds, minus one third, minus two thirds. Or all three with their signs flipped. Check: four ninths plus one ninth plus four ninths is nine ninths, which is one. It passes. Second: ratios minus eighteen, twelve, minus four. Squares: three hundred and twenty-four, one hundred and forty-four, sixteen. They add to four hundred and eighty-four, and the root of that is twenty-two exactly. So the cosines are minus eighteen over twenty-two, twelve over twenty-two, minus four over twenty-two — which tidy to minus nine elevenths, six elevenths, minus two elevenths.

7:49Check: eighty-one plus thirty-six plus four is one hundred and twenty-one, over one hundred and twenty-one. One again. And notice something about that one. Every entry was even, so we could have divided by two first and worked with minus nine, six, minus two instead. The root would have been eleven rather than twenty-two, and the cosines would have come out identical. Cancelling a common factor first is always allowed, because the normalisation was going to kill it anyway.

8:26Where do ratio triples actually come from? Most often, from two points. Take two points on a line. The three coordinate gaps between them are direction ratios of that line. Straight off, no work. And here is a small thing that unsettles people. Which way round do you subtract? Either way. Both are correct. Subtract the first point from the second and you get one triple. Subtract the second from the first and you get every entry negated — which is the original triple multiplied by minus one.

9:05Minus one is a perfectly good value of k. So both triples are ratio triples, and they are ratio triples for the same line. They do not normalise to the same cosine triple, though. They normalise to the two opposite ones — which is precisely the point we made a moment ago. A ratio triple carries a line, not a sense. So if your answer and someone else's differ by a sign throughout, you have not made a mistake. You subtracted the other way round.

9:42Now an application, and a trap inside it. Three points are collinear when they all lie on one line. How do you test that with ratios? Take the gaps from the first point to the second. Take the gaps from the second to the third. If those two triples are proportional — one is a constant multiple of the other — then... Then the two segments are parallel. That is all you have shown.

10:10And parallel is not collinear. Two parallel segments floating in space are two different lines. Proportionality alone cannot tell them apart from one line. What finishes the argument is the shared point. The two segments we chose both contain the middle point. Two parallel lines that share a point are not two lines — they are one line. That is the second half of the test, and it is the half that gets skipped, because it feels too obvious to say out loud. Say it out loud.

10:48Worked: take the points two, three, minus four; one, minus two, three; and three, eight, minus eleven. First gaps: minus one, minus five, seven. Second gaps: two, ten, minus fourteen. Is the second a multiple of the first? Minus two times minus one is two. Minus two times minus five is ten. Minus two times seven is minus fourteen. Yes — proportional, with k equal to minus two. And the middle point is in both segments. Both halves. Collinear.

11:28One more, and it makes a small extra point. The points two, three, four; minus one, minus two, one; and five, eight, seven. This time take the gaps from the first point to the second, and from the first point to the third. First to second: minus three, minus five, minus three. First to third: three, five, three. The exact negatives. So k is minus one — proportional — and the first point is shared. Collinear.

12:00The extra point is this. I picked the first point as the shared one, but I could have picked any of the three and used the other two gaps. The verdict comes out the same every time. Which makes sense: collinear is a property of the three points, not of the order you happened to write them in. Back to the recovery for one last warning, because this one costs marks.

12:28When you normalise a triple, you divide by the square root of the sum of its squares. It is very tempting to read that root as the distance between two points. It is not. It is the length of whichever triple you were handed. Sometimes those coincide. If the triple happens to be the coordinate gaps between two named points, then yes, that root is the distance between them. But the triple usually did not come from two points at all. Two, minus one, minus two came from nowhere in particular; so did minus eighteen, twelve, minus four. There are no points, so there is no distance. The root is just the size of the triple.

13:16And even when there are points, the moment you scale the triple — which you are perfectly entitled to do — the root scales with it and stops being the distance, while the cosines you recover do not change at all. Length of the triple. Not distance between points. Those are two different sentences. Two small facts about zeros, and then we are done. First: an individual entry may certainly be zero. If a is zero, then l is zero, so the first direction angle is a right angle, and the line is perpendicular to the first axis. A zero entry is information, not a problem.

14:02Second: all three cannot be zero at once. The multiplier k is required to be non-zero, and in any case a triple of three zeros has a length of zero, so there is nothing to divide by. It names no line. So: any entry may vanish. Not all of them. To close, a look forward, because ratio triples are not an aside — they are the material the next stage is built out of.

14:34The moment you want to write down the equation of a line in space, you need two things: a point the line passes through, and a direction for it to run in. That direction arrives as a ratio triple, because that is the form directions naturally show up in. So it is worth having the distinction clean before you go on. Direction cosines: a specific triple, fixed length, two per line, squares adding to one.

15:05Direction ratios: an entire family, closed under multiplying by any non-zero number, endlessly many per line, no member special. One multiplier serves all three entries. Divide by the length of the triple to come back, and keep both signs when you do. And proportional ratios give you parallel — you need a shared point to get collinear.