Two equations in two unknowns nearly always have a solution, so finding λ and μ proves nothing. Everything this sub-topic asks is decided by the equation you did not use, and students stop one step before it.
Start on the skew case and cover the right half of the screen. With only the plan visible, ask whether the lines meet. They cross at (1, 2) and the class will say yes. Then uncover the heights: 3 and 2.
That is the sub-topic in one move, and it is worth doing physically with a hand or a sheet of paper rather than describing. A view from above loses exactly one piece of information, and that piece is the answer.
Then press to the intersecting case, where the plan crosses at (3, 1) and both heights are 5. Same method, different outcome, which is the point: the method does not know in advance.
The last two cases have no crossing in the plan at all, which is worth a sentence: when the directions are parallel there is nothing to solve, and the question becomes the much easier one of whether a point of one lies on the other.
| Question | Answer |
|---|---|
| 1. z on L₁ at λ = 0 | 3 + 2(0) = 3. |
| 2. z on L₂ at μ = 1 | 2(1) = 2. |
| 3. The intersection's z | 3 + 2(1) = 5. |
| 4. How many equations | 3, one per component. |
| 5. Why skew needs three dimensions | B. Three equations, two parameters. |
Questions 1 and 2 together are the answer to "are these lines skew", split into two steps so neither can be skipped. A student who gets both and still writes "they intersect" has not understood what the two numbers were for, which is worth catching here rather than in a paper.
1 markTesting whether the directions are parallel, with a reason.
1 markAll three component equations written out.
1 markSolving two for λ and μ.
1 markSubstituting into the third and stating the conclusion in words.
The last mark is for a sentence. "3 ≠ 2" on its own is working; "the equations are inconsistent, so the lines are skew" is the answer. Insist on it in class, because students who write only the numbers lose a mark on work that was entirely correct.
| They wrote | What happened |
|---|---|
| "They intersect", with λ and μ found | The error this page exists for. Stopped after two equations. Cover the right half of the figure instead of explaining. |
| λ = 0 as the answer to question 3 | Gave a parameter where coordinates were asked for. Worth naming: a point of intersection is three numbers. |
| Checked the equation they had already solved | They have substituted back into x or y, which must work. Insist the unused equation is identified in writing before solving. |
| "Parallel" for the skew pair | Mis-tested the directions, usually by comparing only the first components. (1, 2, 2) against (2, −1, 2) needs the same factor three times. |
| "Skew" for the parallel pair | Went straight to the equations without the direction test, found no solution, and concluded skew. Parallel distinct lines also have no solution, which is why the direction question comes first. |
| "Coincident" for the parallel pair | Correct that the directions match, then did not test a point. (0, 0, 0) is not on the first line. |
| 2 on question 1 | Used the other line. Worth checking they know which line is which, because the whole method depends on keeping them apart. |
| 6 on question 4 | Counted three equations per line. The two position vectors are set equal to each other once. |
"How can lines not meet if they are not parallel?" Give them a physical example before any algebra. Hold a pen along the edge of a desk and another pointing away from it a hand's width above: not parallel, never meeting. Then ask them to look down on it from above, where the two appear to cross. The desk demonstration takes ten seconds and makes the algebra a confirmation rather than a surprise.
"Which two equations should I solve?" Whichever pair is easiest, usually the two with the simplest coefficients. It makes no difference to the answer, and saying so matters, because students worry that choosing differently from the teacher will give a different conclusion. It cannot.
"What if the directions are parallel AND the equations have no solution?" Then the lines are parallel and distinct, and the equations were never going to help. That is why the direction test comes first: it saves the work and it prevents the wrong conclusion. A class that goes straight to the simultaneous equations will label parallel lines skew.
"Do we ever find the distance between skew lines?" Not at this level. The guide asks only for the four cases and for points of intersection, and the shortest distance between skew lines is not on the course. Say so clearly, because it appears in textbooks and some students will have seen it and worry.
| Demonstrate | Feed all three equations into the Nspire's system solver and let it return no solution. That message is the answer, not a failure, and seeing the machine say it is oddly convincing. Then change one number so the lines intersect and watch it return λ and μ. |
|---|---|
| Where they stick | The Casio's SIMUL solver handles two to six unknowns, so do not tell them it is limited to two. The real constraint is that it needs as many equations as unknowns: it cannot take three equations in two parameters. So it cannot do the whole question, which is the right division of labour anyway: solve two, check the third by hand. |
| The check | Substituting the parameter back to get the POINT, rather than quoting the parameter. One matrix line on the Nspire, and it heads off the commonest presentation error. |
Worth saying that these questions are nearly always Paper 1 and the numbers are built to be whole, so the calculator is a check on the elimination rather than a method.
| Step | What |
|---|---|
| 1 | The two pens on the desk. Not parallel, not meeting. Then look from above. |
| 2 | The two questions, in order: directions parallel, then share a point. Build the four-case table. |
| 3 | Note that only skew is new, and that it cannot happen in two dimensions. |
| 4 | The skew pair. Direction test first, then all three equations written out. |
| 5 | Solve x and y. Identify the unused equation out loud before substituting. |
| 6 | The figure, with the right half covered, then uncovered. |
| 7 | The intersecting pair, same method, and the point found by substituting back. |
| 8 | Parallel against coincident, settled by testing a point rather than solving. |
Do not say "solve the equations to see if they meet". It is the instruction that produces the error, because solving succeeds and meeting does not follow. Say "solve two and test the third", every time, as a single phrase.
Do not draw skew lines in two dimensions on the board without saying what you are doing. Any flat picture of skew lines is a projection, and for almost every viewing direction a projection of skew lines crosses. (Look along a direction lying in the plane of the two directions and they come out parallel instead, which is the one case that does not cross. Worth knowing, because a student may well draw it.) If you draw one, say that the crossing is an artefact of the view, or the diagram will teach the opposite of the lesson.