Two things go wrong at the start of vectors and they go wrong all term: adding magnitudes, and writing AB the wrong way round. Both are cheap to fix in the first lesson and expensive to fix in the fourth.
Start the slider at 0, where the two readouts agree at 10, and ask whether |a + b| = |a| + |b|. It is true, there, and a class will say so. Then drag it and watch the green bar fall away from the red one while the red one does not move.
The two numbers worth stopping on are 7.0711 at 90°, which is the worked example, and 0 at 180°. The second is the one that settles the argument: two vectors of length 5 can add to nothing at all. No amount of arithmetic on the number 5 produces that.
At 90° the figure shows b as exactly (−4, 3), so the figure and the worked example are the same problem. Point that out; students otherwise treat the figure as a separate illustration.
| Question | Answer |
|---|---|
| 1. |a| | √25 = 5. |
| 2. |a + b| | √50 = 7.0711. |
| 3. |AB| | √26 = 5.099. |
| 4. |3a| | √225 = 15, which is 3 × 5. |
| 5. When lengths add | B. Parallel and the same way. |
Questions 2 and 4 are deliberately next to each other. Adding does not scale the length and multiplying by a scalar does, and a student who has both right has the distinction that the whole sub-topic rests on.
1 markThe displacement as destination minus start.
1 markThe components, correct and in order.
1 markThe magnitude from the sum of squares.
On a proof the marks are for staying in vectors. A student who assigns coordinates and grinds through arithmetic will often reach the right conclusion and score less, because the question was testing the vector method. Say so before setting one.
| They wrote | What happened |
|---|---|
| 10 for |a + b| | The error this page exists for. Added the magnitudes. Slider to 180° rather than an explanation. |
| 7 for |a| | Added the components. Worth catching early: the magnitude is Pythagoras, and 3 + 4 is a perimeter-shaped answer to a length question. |
| 25 for |a| | Forgot the square root. Very common while the notation is new, and it corrects itself quickly. |
| 5 for |3a| | Scaled the vector and not the length, or simply quoted |a|. Ask what 3a is in components first. |
| 45 for |3a| | Applied the 3 twice. They have found |3a| = 3|a| and then multiplied again. |
| 5 for |AB| | Used the two-dimensional pair and dropped the z difference of 1. Worth noticing because it recurs through the whole vector run. |
| AB = a − b | The second error this page exists for. The magnitude comes out right, so it survives every length question and fails every direction one. Insist on saying "destination minus start" aloud. |
| A unit vector whose components do not square to 1 | Divided by the wrong number, usually by a component rather than the magnitude. The check is five seconds and worth demanding every time. |
"Why is it called the triangle inequality?" Because a, b and a + b form a triangle when you draw a and b nose to tail, and one side of a triangle cannot be longer than the other two together. The equality case is the degenerate triangle where all three lie on a line. Drawing that once makes the inequality obvious rather than a fact to learn.
"Does |a − b| have a rule too?" Yes, and it is worth half a minute: |a − b| is the distance between the two tips. For our a and b it is |(7, 1)| = 7.0711, the same as |a + b| here, and perpendicularity alone is what does that: |a + b|² − |a − b|² = 4(a · b), so the two are equal exactly when the scalar product is zero. Equal length is not part of it. (5, 0) and (0, 2) are perpendicular, of different lengths, and both sums come to 5.3852. Students who notice the coincidence are paying attention, and the one-line identity is the answer rather than a list of the things that happen to be true here.
"Which letter is the position vector?" Lower case for the vector, upper case for the point: A is a place and a is the arrow from the origin to it. Exam questions rely on that convention without stating it, so it is worth being pedantic in the first lesson.
"Do I need the k component if everything is flat?" No, and two-dimensional questions appear. But the magnitude formula and every method on this page extend to three dimensions without change, which is one of the real arguments for vectors, so it is worth doing at least one three-dimensional example in the first lesson so nobody thinks of z as an extra topic.
| Demonstrate | Two lines, whichever machine. On the Casio, Norm(VctA+VctB) then Norm(VctA)+Norm(VctB); on the Nspire, norm(a+b) then norm(a)+norm(b), with ctrl enter on the first so it gives 7.0711 rather than the exact 5·√2. Either way the pair reads 7.0711 and 10. The error, on the screen, in two keystrokes. Do it before the explanation, not after. |
|---|---|
| Where they stick | Using Abs for a vector magnitude on the Casio. Abs is for real and complex numbers and will not accept a vector; the command is Norm, under OPTN → MAT/VCT and then paged across with F6. Worth saying once, clearly, because the error message is unhelpful and the submenu is three presses deep. |
| The check | A unit vector's Norm should return exactly 1. If it does not, they divided by the wrong thing, and this catches it instantly. |
Entering a 3×1 vector on the Casio is the step that wastes the most time, because there is no Vector app on the icon menu: it is Run-Matrix, then the MAT/VCT soft key, then set the dimension, fill it, and EXIT. Walk through it once with the whole class rather than letting them hunt the icon menu.
| Step | What |
|---|---|
| 1 | Magnitude and direction as the definition. Two arrows the same length and direction are the same vector. |
| 2 | Components and base vectors. The magnitude as Pythagoras, in two dimensions and then three. |
| 3 | Ask for |a + b| with a and b both of length 5, before any theory. Collect the 10s. |
| 4 | The figure. Slider to 180°, where the answer is 0. |
| 5 | The triangle inequality, drawn nose to tail so the name explains itself. |
| 6 | Scalar multiples, which DO scale the length. Put |3a| next to |a + b| deliberately. |
| 7 | Position against displacement, and AB = b − a said aloud several times. |
| 8 | Unit vectors, with the squares-to-1 check. Then the midpoint proof, in vectors only. |
Do not say "a vector is a line with an arrow on it". It invites the idea that moving the arrow changes it, and then translating a vector in a proof feels illegal. Say a magnitude and a direction, and show two arrows in different places that are the same vector, in the first five minutes.
Do not let AB = a − b pass as a slip. It is the single most persistent error in the vector run, it survives every magnitude question, and by 3.14 it is producing lines that point the wrong way with no symptom. Correct it every single time for the first two lessons and it stops.