Higher Level only, and the first of four sub-topics that build in order: 3.10 vectors, 3.11 lines, 3.12 motion, 3.13 products. Anything shaky here is shaky three times more later.
Press Sweep and say nothing until it has gone out and come back. The green bar rises and falls between 1 and 7; the red bar does not move. Two arrows that never change length produce a resultant that changes by a factor of seven.
Then stop it at 0° and point out that the red bar is finally right. That is why the error survives: adding magnitudes is not nonsense, it is the answer to a different and much rarer question.
If the class is quick, ask before the sweep what the smallest possible resultant is. "Zero" is the commonest guess and it is wrong here, because 3 and 4 are not equal. 1 is the floor, and it is 4 − 3.
| Question | Answer |
|---|---|
| 1. |(3, 4)| | √(9 + 16) = 5. |
| 2. First component of the unit vector | 3/5 = 0.6, and the vector is (0.6, 0.8). |
| 3. 3 N and 4 N at 90° | 5 N, because the two forces are the two components. |
| 4. The same at 120° | Components 1 and 3.46, so √13 = 3.61 N. |
| 5. When |a + b| = |a| + |b| | B. Only when they point the same way. |
Question 3 is deliberately the 3-4-5 triangle, so the arithmetic is free and the only thing being tested is whether they resolved rather than added. Question 4 then removes the friendly numbers and keeps the same method.
1 markResolving into components.
1 markAdding componentwise.
1 markThe magnitude, with units.
An answer of 7 scores zero on a three-mark question, which surprises students who feel they did something. Say it explicitly: no step of adding magnitudes is a step of the method, so there is nothing for a markscheme to award.
| They wrote | What happened |
|---|---|
| 7 on question 1 | Added the components. Same instinct as adding magnitudes, one level down. |
| 25 | Stopped before the square root. Common and cheap to fix; the units give it away, since 25 would be in metres squared. |
| 0.8 on question 2 | Gave the second component. Read the question, not the mathematics. |
| 3 on question 2 | Did not divide at all. Ask them to check the magnitude of their answer: it is 3, not 1. |
| 0.43 | Divided by 7 instead of 5. They have used the sum of the components as the magnitude, which is the question-1 error carried forward. |
| 7 on question 3 or 4 | Magnitudes added instead of vectors. At 120° it is nearly twice the true answer. |
| 6.08 on question 4 | Used cos 120° as +0.5. The sign of the cosine past 90° is the whole difference between 6.08 and 3.61, and it links straight back to the unit circle in 3.8. |
| 1 on question 4 | Treated 120° as directly opposed. They have rounded the geometry to "mostly against". |
"Why is walking 3 km then 4 km seven kilometres, then?" Because distance is a scalar and it is the total path length, which does add. Displacement is the vector from start to finish and it is between 1 and 7. Both numbers are correct answers to different questions, and physics questions are usually asking for the second. This is the single most useful sentence in the sub-topic.
"Is a unit vector the same as a direction?" It is how we write a direction as a number, yes. The useful consequence is that any vector can be rebuilt as magnitude times unit vector, which is exactly how 3.12 writes a velocity: 20 m s⁻¹ in the direction (0.6, 0.8) is (12, 16).
"Which way round is AB?" Destination minus start, b − a. The trap is that getting it backwards does not change the magnitude, so a question asking only for the distance AB forgives it and the next question, asking for a direction, does not. Teach it as a habit before it costs anything.
"Does i, j, k notation matter?" Only as something to read. The syllabus uses columns and base vectors interchangeably, examiners accept either, and the one thing to insist on is that a student does not mix them inside a single line of working.
| Stage | What to do |
|---|---|
| Demonstrate | Show norm(a+b) against norm(a)+norm(b) on one screen, with a and b at 120°. 3.61 and 7.00. The machine makes the distinction visible in two lines, which is faster than any amount of explanation, and it inoculates against the error rather than correcting it afterwards. |
| Where they stick | The Casio brackets, and the command name. The magnitude is Norm(, under OPTN → MAT/VCT; Abs is the absolute value of a real or complex number and will not take a vector. Then the brackets: Norm(Vct A + Vct B) is the resultant, 3.61, while Norm(Vct A) + Norm(Vct B) is 7, and that second one is the error the whole page is about. Both run without complaint. On the Nspire, the dimension mismatch between a 2-vector and a 3-vector is the usual stumble: keep a question in one dimension throughout. |
| The check | Any resultant must lie between the difference and the sum of the magnitudes. For 3 and 4 that is 1 to 7, so an answer outside that range is wrong before it is checked. It is a one-line sanity test that works on every resultant question in the course. |
Degrees mode for every question with a degree symbol, which is nearly all of them here. The resolving step calls cos and sin, so unlike 3.7's sectors the mode does matter.
| Step | What |
|---|---|
| 1 | The walk: 3 km then 4 km. How far have you walked, and how far are you from home? Two answers. |
| 2 | Scalar against vector, from that example, not from a definition. |
| 3 | Figure, Sweep, out and back. Stop at 0°. |
| 4 | Components, addition, magnitude. The 3-4-5 case first. |
| 5 | Unit vectors, with the 0.6² + 0.8² = 1 check done out loud. |
| 6 | Position vectors and AB = b − a, with two named points. |
| 7 | The 1-to-7 bound as a closing habit, then the questions. |
Do not say "a vector is an arrow". It is drawn as one, and the picture is worth having, but a student who believes the arrow is the vector gets stuck the moment two arrows in different places are the same vector. Say that a vector is a size and a direction, and that an arrow is one way of drawing it.
Do not say "just use the calculator for magnitudes". Entering a vector takes longer than squaring two numbers and adding them, and the hand route keeps the components in view, which is where the error actually is. Reach for the machine when the angle is awkward, not for √(9 + 16).