One point plus one direction fixes a line, in vector form and in the symmetric Cartesian form the parameter eliminates to

0:00There are two ways to pin down a line in space, and they sound equally reasonable. One: give a point the line passes through, and give a direction for it to run in. Two: give two points, and take the line through both. Either one is enough. But only the first gets built, in almost every treatment you will meet, and the second is quietly handed to you later as a result.

0:27So here is the plan. We build the first case properly, out of one fact you already have. Then, when we have it, the second case takes two lines and we do it on the spot rather than waiting for it to be given. And the whole thing gets written twice: once with vectors, where it is short, and once in coordinates, where it turns into three equal fractions. Those two look nothing like each other and describe exactly the same set of points.

1:01Start with the ingredients. A fixed point — call it the anchor — and a direction. The anchor has a position vector: the arrow from the origin to it. The direction is just a vector, and here is the thing worth getting right immediately. The direction vector does not have to lie on the line. It does not have to reach from the anchor to anywhere in particular. It only has to be parallel to the line. Draw it off to one side, clear of everything, because the direction is not a piece of the line — it is something the line has.

1:41Now put a moving point on the line and let it move. Its position vector is the thing we want an equation for. One number is going to control that movement. Turn the number up and the point slides one way; turn it down through zero and the point runs back the other way. One dial, one line. The derivation is three lines and it uses one fact you already own.

2:10Line one. The join from the anchor to the moving point lies along the line, and the direction vector is parallel to the line. So the join is parallel to the direction vector. Two parallel vectors means one is a multiple of the other, so the join is some real number times the direction. Line two. That same join is also the difference of two position vectors: the moving point's, minus the anchor's.

2:41Line three. Those are two descriptions of the same arrow, so set them equal and rearrange. The moving position vector equals the anchor's position vector plus that number times the direction. That is the equation. The number in it is called the parameter, and it is almost always written with the Greek letter lambda. Notice what actually did the work. Parallel means multiple. Everything else was bookkeeping. That equation has to be read in two directions, and one of the two usually gets a single sentence.

3:21Forwards: pick any value of lambda, do the arithmetic, and you get a point. Every value gives a point of the line — positive values run one way, negative values run back the other, and lambda equal to zero hands you the anchor itself. Backwards: hand it any point of the line, and there is exactly one value of lambda that produces it. The backwards reading is the one that matters. Without it, the equation would only be a property that some points of the line happen to satisfy. With it, the equation is a description: it generates the whole line and nothing else.

4:05Both halves were checked over seven hundred and two lines and nine dial settings each. Feed a setting in, take the point out, hand it back: the same setting comes out, at all six thousand three hundred and eighteen cases. Write the direction vector by components and something familiar falls out. Those three components are a set of direction ratios for the line. That is the whole hinge between this and the previous topic, and it runs both ways: given direction ratios, that same combination of the three unit vectors is a vector parallel to the line.

4:49So the previous topic's freedom is inherited whole. Any non-zero multiple of the direction vector is just as good, because any proportional triple is just as good. One warning, and it exists because the same letter gets used for two different things on the same line of working. The middle component of the direction vector is a component. The same letter under modulus bars is the length of the whole vector. For three, two, minus eight, the middle component is two, and the length is the square root of seventy-seven, which sits between eight and nine. They are not close, and they are not the same kind of quantity.

5:37Two builds. The first has both ingredients handed to you. Through the point five, two, minus four, parallel to the direction three, two, minus eight. The equation is the anchor's position vector, plus lambda times the direction vector. Substitute and you are finished. There is nothing to simplify and nothing to check. A second one, same shape. Through one, two, three, parallel to three, two, minus two. Same substitution, same one line of work.

6:12Set lambda to one and you step to four, four, one. Set it to minus one and you go the other way, to minus two, zero, five. Both of those are on the line, and both were found by turning one dial. That is genuinely all there is to the vector form when the direction is given. The interesting version is when it is not. Second build. A line through the point one, two, minus four, at right angles to two other lines.

6:46You have the point. You do not have a direction — you have a condition on it. It has to be square to two given directions at once. You could set that up as two equations in three unknowns and solve. Or you could take the cross product, which is a vector at right angles to both its factors in one step. That fact comes from vector algebra, not from anything about lines.

7:14The two given directions are three, minus sixteen, seven, and three, eight, minus five. Cross them and you get twenty-four, thirty-six, seventy-two. Now the previous topic pays for itself. Every entry has a factor of twelve in it, and any proportional triple names the same direction — so throw the twelve away and use two, three, six. It is square to both as well, and it is far easier to carry.

7:46The line is the point one, two, minus four, plus lambda times two, three, six. The cross product being square to both was checked at all six hundred and twenty-four pairs of grid directions that are not parallel. The fifty-two pairs where it fails are exactly the parallel ones, where there is no single perpendicular to find. Here is the thing about this equation that makes marking it awkward. A line does not have an equation. It has endlessly many, and two correct answers can share not one single number.

8:26Two reasons. Any point of the line will do as the anchor — slide the anchor anywhere along the line and the line is unchanged. And any non-zero multiple of the direction will do as the direction — stretch it, shrink it, reverse it, the line is unchanged. Counted: re-anchoring at any of nine points of the line left the line exactly where it was at all six thousand three hundred and eighteen cases, and rescaling the direction by any of seven numbers — three of them negative, two of them fractions — left it there at all four thousand nine hundred and fourteen.

9:09And that is not because the check says yes to everything. Move the anchor to a point that is not on the line, keeping the same direction, and you name a different line every single time — at all seventeen thousand four hundred and twenty-four cases. Which tells you how to settle a disagreement. Do not compare the two answers symbol by symbol. Take a point off one and substitute it into the other.

9:40Now the second way of pinning a line down, the one that gets announced and then skipped. Two points are given. No direction. But the join from the first to the second is a direction for the line — that is the only idea in it. So: subtract the position vectors to get the direction, then use the result we already have, with either point as the anchor. Worked: from one, two, three to four, six, eleven, the join is three, four, eight. Anchor at the first point. Lambda equal to zero returns the first point; lambda equal to one lands exactly on the second. Both named points are on the line, which is the only thing the construction had to deliver.

10:31That is two lines of work, and it is worth doing yourself rather than looking it up, because the version you look up is a formula with two position vectors in it and no reason attached. Now the second half. Everything so far was vectors. We are going to turn it into coordinates. Write all three vectors by components: the moving point, the anchor, and the direction. Substitute into the vector equation.

11:03Now compare the coefficients of the three unit vectors separately. That step is allowed for a specific reason, and it is worth saying: the three unit vectors are independent, so two vectors are equal only when all three pairs of components agree. It is not a convention, it is a fact about the frame. Out come three scalar equations. Each coordinate is its anchor value, plus lambda times the matching direction entry.

11:36These are the parametric equations, and the whole content is sitting in plain sight: the same lambda appears in all three. One dial, three coupled readouts. Turn it once and all three coordinates move together. Which raises the obvious question. What if you let the three have their own parameters? Then you are not describing a line any more. You are describing a solid region of space. Here is the size of the mistake. Take an anchor and a direction, and nine settings for the dial. One dial gives nine points, strung out along a line. Three independent dials give seven hundred and twenty-nine — a whole block of space.

12:26Across seven hundred and two lines, three dials sweep more than one dial does at five hundred and forty of them. The hundred and sixty-two where they do not are exactly the lines drawn parallel to an axis — where two of the three dials have nothing to multiply, so there is nothing extra for them to sweep. So when you see the same letter three times, that is not repetition. That is the constraint.

12:56Now the step that looks like tidying up and is not. Take each of the three parametric equations and make lambda the subject. Each one gives lambda as a coordinate gap over the matching direction entry. Three expressions, all equal to the same lambda, so all equal to each other. Write them in a row joined by equals signs and you have the Cartesian form of the line. Something has changed, and it is not cosmetic. Before the elimination, the equations were a recipe: feed a number in, take a point out. After it, there is no number to feed in. What you have is a condition — these three particular ratios agree — and a condition is something you can hand any point to and get a yes or a no.

13:50That was checked point by point: over eighty-seven thousand seven hundred and fifty pairings of a line with a place, the three quotients agreeing and a dial setting existing gave the same verdict every single time. And lambda is not gone. It survives as the common value of the three fractions. Work out what they all equal, and that number is the parameter — which is how you get a point back out of a written equation without redoing the algebra.

14:25So now you can read one of these backwards, and this is where the marks actually go. The denominators are direction ratios. Read them straight off — no normalising, no tidying, they are a direction and any proportional triple would have done. The numerators are where it goes wrong. The numbers on top are the anchor's coordinates with their signs reversed. So a minus on top means a positive coordinate, and a plus on top means a negative one. That is not a rule to memorise, it is just what x minus x-one looks like when x-one is negative.

15:07Watch it happen. Through five, two, minus four with direction three, two, minus eight: the three fractions are x minus five over three, y minus two over two, and z PLUS four over minus eight. The third one carries a plus because the coordinate is negative. Two of them at once: through minus two, four, minus five, parallel to a line with denominators three, five, six. Two of the three numerators carry a plus.

15:41And the other way round: numerators x minus five, y plus four, z minus six mean the anchor is five, minus four, six. Not plus four. That single sign is the whole content of the question. There is one case nobody warns you about, and then you get asked for it. What if a direction entry is zero? Then one of the denominators is zero, and you have written down a fraction that does not mean anything.

16:14It is not an error. It is a condition. A zero direction entry says that coordinate never changes — the line is confined to a plane where that coordinate is fixed at the anchor's value. Checked at every line on the grid whose direction has a zero in it — four hundred and eighty-six of them — and at every single one, every point of the line carries the anchor's coordinate in that slot.

16:44The sharpest case: the line along the first axis through the origin. Its direction is one, zero, zero, so TWO of the three denominators vanish. Written as fractions it is nonsense. Written as conditions it is completely clear: the second and third coordinates are both zero, and the first runs free. And in vector form there is no problem at all — it is just lambda times the first unit vector. When denominators vanish, reach for the vector form and say the conditions out loud.

17:24One more trap, and it is tempting because it looks like simplifying. The Cartesian form is fractions, so clear them. Cross-multiply and get one equation with no denominators in it. That equation is not the line. A single equation in three coordinates is a plane. Checked at all seven hundred and two lines, the solutions of one cleared equation always outnumber the points of the line — on the first axis through the origin, twenty-five places against five.

18:01A line in space is where two conditions meet, and you need both. Cross-multiply two of the three pairs and, when no denominator is zero, that is enough — at all two hundred and sixteen such lines it picks out the line exactly. But when a denominator does vanish, two is not enough. Two adjacent pairs work at only two hundred and seventy of four hundred and eighty-six such lines; at the other two hundred and sixteen they let in points that are not on the line. Take all three pairings and it is right everywhere.

18:40Which is one more reason to leave the fractions alone. Two shapes, one object. Where does each one earn its keep? The vector form is a generator. It hands you points. Angles and distances come out of it with less writing, and it never produces a denominator that vanishes. The Cartesian form is a test. Hand it a point and it says yes or no, faster than solving for a parameter. And you can read a point and a direction straight off it, if you remember to flip the signs.

19:17One more variant you will see. Put the direction COSINES under the fractions instead of the ratios. It is the same line — cosines are a set of ratios like any other. What changes is the meaning of the common value. With cosines underneath, the direction has length one, so the common value is the actual distance travelled from the anchor. Signed: forwards is positive, backwards is negative. That was checked at all two thousand eight hundred and eight forward cases.

19:53With ordinary ratios underneath, it is not a distance — it agrees with one at only six hundred and forty-eight of those same cases, the ones where the direction already happened to have length one. So: one point, one direction, one dial. Turn the dial and you sweep the line; eliminate it and you get a test. Same line, either way.