Two lines can cross when you look down on them and still miss each other completely. Solve the x and y equations and they agree at (1, 2); check z there and one line is at 3 and the other at 2. Directly above that spot on the plan one line sits one unit higher than the other, so they never touch.
The left half is a view from above, where height is invisible. The right half is the height. Both are needed, and the left half on its own is what makes skew lines surprising.
Everything here comes from two yes-or-no questions, asked in this order:
| Directions parallel? | Share a point? | The lines are |
|---|---|---|
| Yes | Yes | coincident: the same line twice |
| Yes | No | parallel and distinct |
| No | Yes | intersecting |
| No | No | skew |
Two questions, four answers, and nothing else to remember. Ask them in that order, because the first is much quicker and it settles half the cases.
Only the bottom row is new. Coincident, parallel and intersecting all exist in two dimensions; skew does not, and cannot, because two non-parallel lines in a plane always meet. Skew is the case that three dimensions add, which is why it appears here and not before.
L₁: r = (1, 2, 3) + λ(2, −1, 2) and L₂: r = (0, 0, 0) + μ(1, 2, 2)
Three equations, two unknowns. Two of them pin the parameters down and the third is then either satisfied or not, with no freedom left. In two dimensions there is no third equation, which is exactly why two non-parallel lines there always meet.
Always state which equation you did not use for solving, and then substitute into it. A student who solves x and y and then checks x has checked nothing. The unused equation is the whole test.
The intersecting case, for contrast. Change L₂ to r = (3, 1, 5) + μ(1, 2, 2) and run the same method. The x and y equations give λ = 1 and μ = 0. Then z on L₁ is 3 + 2(1) = 5 and z on L₂ is 5. They agree, so the lines meet, and the point of intersection is (3, 1, 5).
Note that you get the point by substituting your parameter back into either line, not by writing down the parameters. A question asking for the point of intersection wants coordinates, and λ = 1 is not an answer to it.
If the directions are parallel, the second question is easier than solving anything: take a point from one line and test whether it lies on the other.
r = (1, 2, 3) + λ(2, −1, 2) and r = (3, 1, 5) + μ(4, −2, 4): the directions are parallel, and (3, 1, 5) is on the first line, at λ = 1. So they are coincident, the same line written two ways, which is 3.14 again.
But r = (1, 2, 3) + λ(2, −1, 2) and r = (0, 0, 0) + μ(2, −1, 2) have the same direction and (0, 0, 0) is not on the first line: putting x = 0 needs λ = −½, which gives y = 2.5, not 0. So these are parallel and distinct.
The machine will solve a system for you, and the useful part is that it reports no solution when the system is inconsistent, which is precisely the skew answer. That is worth seeing once, because it reframes "no solution" as information rather than failure.
When you may use it. Analysis Paper 1 is non-calculator, and these questions are nearly always Paper 1, because the numbers are built to be whole. Do the elimination by hand and use the machine to check.
The mark people lose. Solving two equations and concluding the lines meet. Two equations in two unknowns almost always have a solution, so finding λ and μ proves nothing at all. The habit: write down the third equation before you solve anything, so it is on the page waiting to be tested rather than forgotten.
Throughout: L₁ is r = (1, 2, 3) + λ(2, −1, 2).
1. With L₂: r = (0, 0, 0) + μ(1, 2, 2), the x and y equations give λ = 0. Find z on L₁ there.
2. With the same pair, μ = 1. Find z on L₂ there.
3. Now take L₂: r = (3, 1, 5) + μ(1, 2, 2). These two meet. Give the z-coordinate of the point of intersection.
4. How many equations do you get from setting two position vectors in three dimensions equal?
5. Why can two non-parallel lines fail to meet in three dimensions but never in two?
1 markTesting whether the directions are parallel, with a reason.
1 markSetting up all three component equations.
1 markSolving two of them for λ and μ.
1 markSubstituting into the third and stating the conclusion.
The last mark needs the conclusion in words: "the equations are inconsistent, so the lines are skew". A pair of numbers that do not match is the working, not the answer, and an examiner cannot award a conclusion that is not written down.
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