The angle between two lines, from cosines or from ratios

0:00Here are two lines in space. They are not parallel, and they never meet — they pass each other at different heights. So what is the angle between them? That is a real question, not a rhetorical one. An angle is something you measure at a corner, and these two lines do not share a corner. There is nowhere to put the protractor. The answer arrives in three lines, in a small note that most treatments print without comment, and everything else in this topic depends on it.

0:36Here is the construction. Pick any point you like. Through that point, draw a line parallel to the first. Through the same point, draw a line parallel to the second. Those two copies do meet — they both pass through the point you picked. Measure the angle there. That is the angle between the original two lines, by definition. Two things make it work. Parallel lines share direction ratios, so a copy carries the direction of the original unchanged. And the answer does not depend on which point you picked, because sliding a line without turning it does not change its direction.

1:19So the whole of the rest of this topic is really about directions, not about lines. The lines may be anywhere. Only their two direction triples enter the formula. Now put both copies through the origin, and the formula is ten seconds away. Take a point on the first line and a point on the second. The two segments from the origin are vectors, and their components are exactly the two direction-ratio triples.

1:52So the angle between the two lines is the angle between two vectors whose components you already have, and there is a standard result for that: the dot product is the product of the two lengths times the cosine of the angle between them. Rearrange, and the cosine is the dot product over the product of the lengths. Written out: the sum of the three products of matching entries, over the length of the first triple times the length of the second.

2:25That is the whole formula. Nothing in it is new — the only new step was reading a direction-ratio triple as a set of components, and that came from the previous topic. Except the formula, as it is always written, has modulus bars around the entire quotient. And those bars usually go unexplained, which turns a one-line result into a spell. Here is what they are for. A line has two directions. Nothing in the problem says which one you should take. So when you write down a triple for a line, you have quietly made a choice, and the other choice would have given you the negatives of all three entries.

3:14Flip that choice and the numerator flips sign. The denominator is a product of two lengths, so it does not. The whole quotient comes out negated. The two answers you can get are supplementary — one acute, one obtuse, adding to a straight angle. They are the two angles at a crossing, and they are both there in the picture. The bars pick the acute one, every time, and throw away a choice you never really made.

3:46Checked on a grid of fifteen thousand three hundred and seventy-six pairs of lines: before the bars the quotient comes out negative at six thousand six hundred and fifty-six of them, more than two in five. After the bars, at none of them. And reversing either line negates the raw quotient at every single pair, while leaving the barred one exactly where it was. So the bars are not tidying — they are discarding information that was never about the lines.

4:20One consequence worth stating. If a question asks for the obtuse angle, the formula will not give it to you. Compute the acute one and subtract from a straight angle. There is a second version of the formula, and it is the same formula with a piece missing. Suppose the two triples are direction cosines rather than direction ratios — that is, suppose they have already been normalised. Then each one squares to one. So each length is one. So the denominator is one times one.

4:59The formula collapses to the modulus of the sum of the three products, with nothing underneath at all. That is the identity from the first topic doing its second piece of work here, and it is worth noticing rather than memorising a second formula. Checked at all hundred and twenty-four live triples on the grid: normalise and the squares total one. And the bare sum of products agrees with the full quotient at every one of the fifteen thousand three hundred and seventy-six pairs.

5:36There is also a companion formula for the sine, and it is worth about two minutes — partly because it is not a new result, and because the version usually written down for cosines carries a sign that cannot be right. The sine version puts three brackets under a root. Each bracket is a two-by-two combination: one entry of the first triple times another entry of the second, minus the other way round.

6:08Where does it come from? Not from anywhere new. The sum of those three squared brackets is exactly the product of the two summed squares, less the square of the numerator you already have. That holds at all fifteen thousand three hundred and seventy-six pairs. It is the cosine formula rearranged, nothing more. Now the sign. The version written for cosines commonly carries a MINUS between the first and the second bracket, where the version for ratios has a plus.

6:45That cannot be right. Cosines are ratios that happen to be normalised, so the second formula has to be the first with the denominators set to one, and the first has all three signs positive. And it is checkable. With a minus in that place, the expression under the root goes negative at three thousand one hundred and fifty-two pairs of the grid. A sine cannot be the root of a negative number. One triple along the first axis and one along the third gives minus one under the root.

7:23So: plus, in all three places. And do not memorise it. The sine formula gets used for exactly one thing, which is the next scene but one. Two special cases, and they come from the two formulas — one each. Perpendicular first. Two lines are at right angles when the cosine is zero. Now look at what the cosine is. A numerator over a denominator, and the denominator is a product of two lengths.

7:56A length is zero only for a triple of three zeros, which is not a direction. So for any two genuine lines, the denominator is strictly positive. It can never be the reason the quotient vanishes. Which means perpendicularity is the numerator alone. The sum of the three products of matching entries is zero. Full stop — no roots, no division. That was counted both ways round: the numerator vanishes at two thousand and sixty-four pairs of the grid, the whole quotient vanishes at two thousand and sixty-four, and they are the same two thousand and sixty-four.

8:41Two one-line examples. Two, five, minus four against three, two, four: six plus ten minus sixteen is zero. Seven, minus five, one against one, two, three: seven minus ten plus three is zero. Parallel is the other case, and it comes from the sine. Two lines are parallel when the angle is zero, so when the sine is zero, so when all three of those brackets vanish. Which happens exactly when the three ratios of matching entries agree.

9:18And here is the misreading. Agreeing in ratio is not the same as being equal. The clearest case: minus two, minus four, minus four against two, four, four. Those are exact negatives. Not one entry is shared. And the two lines are parallel, because each ratio is minus one. On the grid, both readings — the ratios agreeing, and the three brackets vanishing — pick out the same three hundred and fifty-two pairs. At ninety-six of those, the two triples share not one single entry.

9:57So if you are checking parallelism by looking for matching numbers, you will miss most of the cases. Now the arithmetic, and it is the same arithmetic every time. Two lines given in vector form, with directions one, two, two and three, two, six. Numerator: three plus four plus twelve, which is nineteen. The two lengths: one plus four plus four is nine, whose root is three. Nine plus four plus thirty-six is forty-nine, whose root is seven. Both whole, which is not an accident — the pair was chosen that way.

10:39So the cosine is nineteen over twenty-one. And an exercise item later hands you the same two lines with the order swapped, and expects the same answer. It gets it — because which line you call the first does not enter the formula anywhere. In Cartesian form nothing changes except where you read the triples from. They are the denominators. Three, five, four against one, one, two. Numerator: three plus five plus eight, which is sixteen.

11:14Lengths: the root of nine plus twenty-five plus sixteen, which is the root of fifty. And the root of one plus one plus four, which is the root of six. So the cosine is sixteen over the root of three hundred. Do not stop there — the simplification is two steps and it is the part students skip. Three hundred is a hundred times three, so its root is ten root three. Sixteen over ten root three is eight over five root three, and rationalising gives eight root three over fifteen.

11:53A different exercise item, with directions one, minus one, minus two and three, minus five, minus four, lands on that exact same number by a different route. And the cleanest one in the whole topic: two, two, one against four, one, eight. Eight plus two plus eight is eighteen; the lengths are three and nine; eighteen over twenty-seven is two thirds. Now the hardest kind of question in this topic, and almost none of the difficulty is in the angle.

12:29An unknown sits in one of the denominators of each line, and you are told the two lines are perpendicular. Find the unknown. The trap is at the very first step. You cannot read the ratios off the equations as written, because a numerator may carry a coefficient, and the running variable may have been subtracted from the constant instead of the other way round. Rearrange both equations into the standard shape first. Do that, and the ratios come out as minus three, two-sevenths of the unknown, two; and minus three-sevenths of the unknown, one, minus five.

13:13Now the condition, which is one line. Nine-sevenths of the unknown, plus two-sevenths of the unknown, minus ten, equals zero. So eleven-sevenths of the unknown is ten, and the unknown is seventy over eleven. A simpler version of the same question needs no rearranging at all: minus three, twice the unknown, two against three times the unknown, one, minus five. Minus nine of the unknown plus two of the unknown is minus seven of the unknown, minus ten, equals zero, so the unknown is minus ten sevenths.

13:56Both were checked by substituting the answer back and confirming the sum of products really is zero. One last item, and it is the best one in the topic, because no arithmetic could settle it. The first line has direction ratios given as three letters — call them a, b and c. The second has each letter minus the next one round: b minus c, then c minus a, then a minus b.

14:27Show the two lines are perpendicular. There are no numbers to substitute. All you can do is expand the sum of the three products. First, a times b minus c gives the a b term minus the a c term. Second, b times c minus a gives the b c term minus the a b term. Third, c times a minus b gives the a c term minus the b c term.

14:55Six terms, and they cancel in pairs. The two a b terms kill each other. So do the two a c terms, and so do the two b c terms. Zero. So the two lines are at right angles for every choice of the three letters. Not for the ones in the question — for all of them. Checked at all three hundred and forty-three triples of letters from minus three to three, of which three hundred and thirty-six give two genuine lines.

15:27That is what a condition looks like when it is doing something arithmetic cannot. So, what is worth carrying out of this. One construction: two lines anywhere in space get an angle by drawing parallels to both through one point. That is what lets the formula apply to lines that never meet. One formula: the sum of the three products of matching entries, over the product of the two lengths, inside modulus bars. If the triples are already normalised, the denominator is one and vanishes.

16:07The bars mean acute. They are throwing away which direction you happened to pick along each line. If you want the obtuse angle, take it from a straight angle afterwards. Perpendicular is the numerator alone going to zero — the denominator can never do it. Parallel is the three ratios agreeing, which is not the three entries being equal. And the sine formula is not a second result. It is the cosine formula rearranged, its only job is to hand you the parallel condition, and the version usually written for cosines has a sign in it that cannot be right.