Three angles with the axes, and why their cosines square up to one

0:00Here is a question with a very short answer and a slightly longer story. How do you say which way something points in space? On a flat page, one number does it. An angle from a chosen direction, and you are done. In space, one number is not enough, and the standard answer is to use three. Put the tail of an arrow at the origin, where the three axes meet, and let it point wherever you like.

0:29Now look at the angle between that arrow and the first axis. Call it alpha. The angle between the arrow and the second axis: beta. The angle between the arrow and the third: gamma. Three angles, one for each positive axis. They are called the direction angles of the arrow, and every one of them is measured from the arrow to the axis itself, not to the plane it lies in.

0:58That last point is worth holding on to. Gamma is not how far the arrow leans out of the flat ground. It is the angle the arrow makes with the upright axis. Angles are awkward to compute with. Their cosines are not. So take the cosine of each of the three, in axis order, and give them names: l for the cosine of alpha, m for the cosine of beta, n for the cosine of gamma.

1:29Those three numbers are the direction cosines of the arrow, and from here on they do all the work. Notice at once that they are not free to be anything at all. Each of them is a cosine, so each one lies between minus one and one. No direction cosine is ever bigger than one, and if you ever compute one that is, something has gone wrong upstream. But the real constraint on them is much tighter than that, and it is the whole subject of this video.

2:05First, one thing that has to be settled before the numbers mean anything: which way along the line is the arrow pointing? Turn the arrow around, so it points the opposite way along the same straight line, and look at what happens to the three angles. Alpha was the angle to the first axis. Reversed, that angle becomes a hundred and eighty degrees minus alpha. Its supplement. And the same for beta, and the same for gamma. All three become their own supplements, and they do it at the same instant, because there is no way to turn an arrow halfway round.

2:47Now, the cosine of a supplement is the cosine negated. So every one of the three cosines changes sign. Every one. Together. That is the part worth stressing, because the commonest slip here is to imagine that reversing a line flips one of the signs. It cannot. The three signs are locked to each other, and they move as a set. Which leaves a small bookkeeping question. A straight line in space runs both ways. It has no built-in sense of travel. So a bare line offers you two triples of direction cosines, and they are the negatives of each other.

3:32Commit to a direction along that line — decide which way is forward — and exactly one of the two triples is yours. So: a bare line, two triples. A directed line, one triple. That is not a defect in the definition. It is the definition doing its job. The three numbers describe a direction, and a bare line does not have one direction, it has two. Next, an objection that comes up every single time, and deserves a slow answer rather than a fast one.

4:08Everything so far has had the arrow starting at the origin. Almost no line in a real problem starts at the origin. So what are the direction cosines of a line that misses the origin completely? Exactly the same three numbers. Here is why. Take your line, wherever it is. Slide a copy of it, without turning it at all, until the copy passes through the origin. The copy is parallel to the original.

4:41Parallel lines make equal angles with any given axis — that is what parallel means. So the copy opens the same alpha, the same beta and the same gamma as the line you started with. Read the three cosines off the copy. They belong to the original just as much. Direction cosines are a property of direction alone. Not of position, and not, as we will see shortly, of length either.

5:13Now the result this whole video is built around. For any directed line at all, the three direction cosines satisfy one equation. Square them, add the three squares, and the total is one. Always. Exactly one. L squared plus m squared plus n squared equals one. It is a small-looking line, and it is doing enormous work. It says that the three angles are not independent measurements of three separate things. They are three views of one direction, tied together.

5:51Here is the odd thing about this identity. It is very often used before it is ever earned. It turns up mid-argument, with the word "but" in front of it, gets used immediately to pin down some scale factor, and is never proved anywhere. A result you have only ever seen used is a result you will misremember. So let us pay for it properly. The whole cost is one short argument, and every ingredient of it is something you already have.

6:26Start with a directed line and two points on it. Call the first point P and the second point Q, and let Q be the one the line points towards. The direction cosines of that line can be written down immediately, and this is the expression to remember: each coordinate gap, divided by the length of the segment. So l is the gap in the first coordinate over the length. M is the gap in the second coordinate over the same length. N is the gap in the third over the same length again.

7:03And the length is the square root of the three squared gaps added together. Now watch. Square the first quotient: you get the first squared gap, over the squared length. Square the second: the second squared gap, over the same squared length. Same for the third. Add them. The three tops add to the sum of the three squared gaps. The bottom is the sum of the three squared gaps.

7:31The numerator collapses into the denominator, and what is left is one. That is the whole proof. Not a trick, not a lemma from elsewhere — the length was built out of exactly those three gaps, so putting them back over it has to give one. Which leaves one loose end. Where did that expression come from? Why is the cosine of gamma the vertical gap over the length? This is worth doing in full, because the standard argument leaves the key step to a picture and never says it out loud.

8:09Take the two points P and Q. Drop a perpendicular from each of them straight down onto the flat ground plane. Now drop one more perpendicular, from P across onto the upright line under Q. That gives you a triangle with a right angle in it. One leg lies flat. The other leg runs straight up, and its length is exactly the gap in the third coordinate. The hypotenuse is the segment from P to Q.

8:41In that triangle, the cosine of the angle at Q is the upright leg over the hypotenuse — the third gap over the length. Which is what we wanted. But only if that angle at Q really is gamma, and that is the step usually skipped. It is gamma for one reason: the upright leg of the triangle is parallel to the third axis. Two parallel lines, cut by the same segment, make equal angles with it. So the angle inside the triangle equals the angle the line opens with the axis itself.

9:20Say the word parallel out loud when you do this. It is the only thing holding the argument up. Let us test the identity on the cheapest possible case, and it is worth doing because most people never do it. Take the first axis and treat it as a directed line. What angle does it make with itself? Nothing at all. Zero. And with each of the other two axes? A right angle, in both cases.

9:51So its direction cosines are the cosine of zero, the cosine of ninety, the cosine of ninety. One, nothing, nothing. Check: one squared plus nothing plus nothing is one. It passes. The second axis, by exactly the same reading, gives nothing, one, nothing. The third gives nothing, nothing, one. And this is also the answer to another common worry. Can a direction cosine ever equal one? Yes — but only along an axis, and when it does, the identity forces the other two to be nothing. There is no room left for them.

10:34Now a worked case in the direction most questions run: angles in, cosines out. A directed line makes ninety degrees with the first axis, sixty degrees with the second, and thirty degrees with the third. Find its direction cosines. This is just three cosines. Cosine of ninety is nothing. Cosine of sixty is one half. Cosine of thirty is half the square root of three. So the triple is nothing, one half, root three over two.

11:10And now do the thing the question does not ask you to do. Check it. Nothing squared is nothing. One half squared is a quarter. Root three over two, squared, is three quarters. Nothing, plus a quarter, plus three quarters. One. That is the first place the identity earns its keep — as a free check on an answer you have just written down. Change the angles slightly and something appears that alarms people.

11:41Ninety degrees with the first axis, a hundred and thirty-five with the second, forty-five with the third. Cosine of ninety is nothing. Cosine of forty-five is one over root two. And cosine of a hundred and thirty-five is minus one over root two. A negative direction cosine. That is not a mistake, and it is not something to be tidied away. An angle bigger than a right angle has a negative cosine. That is all that has happened. The line leans away from the positive second axis rather than towards it, and the minus sign is the only thing recording that fact.

12:27Strip the signs off and you throw away which way the line points. You would be back to a bare line with two directions and no way to tell them apart. Check the identity anyway: nothing, plus a half, plus a half. One. The minus sign disappears under the squaring, which is exactly why the identity cannot tell you the signs, and why you have to keep them yourself. Here is a question that looks like nothing and catches almost everybody.

13:02A directed line makes equal angles with all three axes. Find its direction cosines. Equal angles means equal cosines. Call the common value c. So the triple is c, c, c. Now use the identity, which is the only fact we have. C squared plus c squared plus c squared is one. Three c squared equals one. So c squared is a third. And here is the trap. C squared is a third. Not c. C is the square root of a third — one over root three, which is about zero point five seven seven.

13:43Writing a third is the single commonest wrong answer to this question, and it comes from dividing by three instead of taking the square root afterwards. Check it against the identity and it collapses at once: three thirds squared add up to a third, not to one. One more thing. A square root has two signs. So the answer is plus or minus one over root three — and, by everything we said about reversing, the sign is the same for all three. Two triples, one line, and you must not mix them.

14:21For the record, the angle itself is a little under fifty-five degrees. The identity has a second use, and it is the more powerful one. Run it backwards. Hand me three numbers and I can tell you instantly whether they can possibly be the direction cosines of anything. Square them, add, and look. One? Then yes, they name a direction in space. Anything else? Then no. There is no line anywhere with those three angles, and the question was broken before you started.

14:58This is a strong filter. Most triples of plausible-looking numbers fail it. Take the three numbers a half, a half, a half: the squares add to three quarters, and nothing points that way. It also means the three numbers carry less freedom than they appear to. Fix any two of them, and the third is settled up to its sign: its square is one minus the other two squares. Three angles. Two of them plus a sign is already everything. That is the identity, read as a constraint rather than as a formula.

15:38Two last worked cases, both of them the same move. First, the line through the point minus two, four, minus five and the point one, two, three, taken in that order. Gaps: one minus minus two is three. Two minus four is minus two. Three minus minus five is eight. Squares: nine, four, sixty-four. They add to seventy-seven. So the length is root seventy-seven, and the direction cosines are three over root seventy-seven, minus two over root seventy-seven, and eight over root seventy-seven.

16:19Check: nine plus four plus sixty-four, all over seventy-seven. One. As it must be. Second, a triangle with vertices at three, five, minus four; at minus one, one, two; and at minus five, minus five, minus two. Find the direction cosines of its three sides. Same move, three times. The first side has gaps minus four, minus four, six, and squared length sixty-eight. The second has gaps minus four, minus six, minus four, and squared length sixty-eight again. The third has gaps eight, ten, minus two, and squared length one hundred and sixty-eight.

17:05Divide each triple by its own length and you have your answer. And notice something the question never mentions. Two of those squared lengths are the same. The triangle is isosceles, and you found that out for free on the way past. So, what have we actually got. A directed line through the origin opens three angles with the three positive axes, and the cosines of those three angles are its direction cosines.

17:38Reverse the line and all three change sign at once. A bare line therefore carries two triples; a directed one carries a single triple. A line that misses the origin is handled by sliding a parallel copy onto it. For a line through two known points, each coordinate gap over the length of the segment gives you the three numbers directly, and that expression is where the identity comes from — the length is built out of those very gaps, so squaring and adding hands the sum back to itself.

18:17And the identity itself: three squares, adding to one. Use it forwards as a check on every answer you write. Use it backwards to throw out triples that cannot exist. Use it sideways to recover a third cosine from the other two. Three angles, one direction, one equation holding them together.