Every class has a student who rewrites correct work because it does not match the answer at the back. The Cartesian form has no parameter in it, so two equations for the same line reduce to the identical thing, and the question is settled in two lines.
Press through all three equations and ask what changed. The answer is: the hollow circle moved, and the step length changed, and the line did not. That is the whole sub-topic in one observation.
Then set the slider to 1 and press through again. The same λ does not land on the same point: the first equation gives (3, 1) and the other two both give (5, 0). And the first equation reaches (5, 0) as well, at λ = 2. So one point carries three different parameter values across the three equations. Students who see that stop thinking of λ as a label attached to a point.
The figure draws the line from the Cartesian form every time rather than from the vector equation, which is why it cannot drift between views. Worth mentioning if a sharp student asks how you know it really is the same line.
| Question | Answer |
|---|---|
| 1. z at λ = 2 | 3 + 2(2) = 7. |
| 2. The speed | |(2, −1, 2)| = 3. |
| 3. The angle between the lines | 63.61°. |
| 4. Distance in 4 seconds | 3 × 4 = 12. |
| 5. Is the classmate right? | B. Yes, on both counts. |
Question 5 is the one to discuss as a class rather than mark. Every wrong option is a belief somebody in the room holds, and option C in particular, that a longer direction vector makes a different line, is worth hearing argued and then settled.
1 markA correct point on the line.
1 markA correct direction vector.
1 markThe equation assembled as r = a + λb.
Tell them explicitly that any valid point and any valid direction earn the first two. Students who believe there is one right answer waste time hunting for it, and some of them cross out work that was already complete.
| They wrote | What happened |
|---|---|
| Correct work, crossed out | The error this page exists for. Their equation did not match the book's. Teach the Cartesian check and this stops. |
| 9 for the speed | Forgot the square root. |b|² is the sum of squares; the speed is its root. |
| 6 for the speed | Used a doubled direction vector. Correct for THAT equation, wrong for this one, and worth distinguishing: as geometry the two are the same line, as kinematics they are different motions. |
| 116.39° on question 3 | Took the obtuse angle, from a reversed direction. Both come from the same two lines, so the answer is the acute one. The fix is the modulus on the scalar product. |
| 26.39° on question 3 | 90 minus the answer, which is the angle to a PLANE with that normal. They have anticipated 3.18 by one sub-topic. |
| 3 on question 4 | Gave the speed rather than the distance. Multiply by the time. |
| A Cartesian form with the wrong signs | The denominators are the direction components and the numerators subtract the point's. A direction of −1 gives a denominator of −1, and students tidy that away. |
| A direction vector with no λ | They have found both ingredients and not assembled them. It is the third mark and it is cheap. |
"How do I know if my line is the same as theirs?" Two checks, and both are quick. Is their point on my line? Substitute it into the Cartesian form. Is their direction a multiple of mine? Divide component by component and see whether you get the same ratio three times, or take the cross product and see whether it vanishes. If both, the lines are identical.
"Which point should I use?" Whichever gives the simplest numbers, and in a question that names a point, that one. If you are deriving a line through A and B, use a as the point and b − a as the direction, and keep the order consistent so a sign error does not creep in.
"Does the angle between lines need them to meet?" No, and that is worth stating now because 3.15 is about lines that do not meet. The angle is between the directions, and directions do not care where the lines are. Two skew lines have a perfectly well-defined angle between them.
"Can λ be negative?" Yes, and it must be: the line extends both ways and negative λ covers the half behind the point a. In a kinematics question negative t may be excluded by the context, and that is a modelling restriction rather than a mathematical one. Students sometimes drop half a line because they think a parameter has to be positive.
| Demonstrate | The multiple test with a cross product: CrossP(VctB,VctC) returns the zero vector when two directions are parallel. That is a one-line answer to "is their direction the same as mine" and it is worth showing before 3.16 formally introduces the operation. |
|---|---|
| Where they stick | The Casio's Angle command returns the obtuse angle when the directions are opposed, and students report it without noticing. Say that the answer must be acute and subtract from 180 if it is not. |
| The check | Evaluate VctA+2VctB and compare with the coordinates they computed by hand. A sign error in one component is invisible in the final answer to many questions and shows up immediately here. |
Storing the point and the direction separately, rather than retyping, is what makes the three-equations comparison quick enough to do in class.
| Step | What |
|---|---|
| 1 | A point and a direction determine a line. Draw it, name a and b on the diagram. |
| 2 | r = a + λb, and the table of points for λ = 0, 1, 2, −1. |
| 3 | Ask whether (3, 1, 5) could be used as a instead. Let them realise it can. |
| 4 | The figure, all three equations. Collect what changed and what did not. |
| 5 | Parametric, then Cartesian. Cartesian has no parameter, which is its advantage, but say plainly that it is still NOT unique: it keeps the point and the scale you started from, so two Cartesian forms of one line need not match. Otherwise they will use string matching as the test. |
| 6 | The two-check method for "is this the same line". |
| 7 | The angle, from directions only, with the modulus and the acute answer. |
| 8 | Kinematics: λ as t, b as velocity, |b| as speed. Then the remark that doubling b changes the motion and not the line. |
Do not say "the equation of the line is ...". Say an equation. The definite article is where the whole misconception starts, and it is a one-word change that costs nothing. Mark schemes say "or equivalent" for exactly this reason.
Do not introduce the angle between lines as though the lines must cross. Students build that assumption from the two-dimensional case, where non-parallel lines always meet, and then 3.15 arrives and they cannot see how skew lines have an angle. Define it as the angle between the directions from the start.