Higher Level only, and the last of the vector run. Both products are new names for arithmetic the class can already do, so the teaching load is entirely in choosing which one a question wants.
Sweep it once, and watch the two bars rather than the vectors. The dot product's bar shrinks to nothing at 90° and then grows downwards through the axis; the area bar is at its tallest at the same instant. They trade places.
Stop it at 90° and ask what the dot product is telling you. Someone will say "nothing". It is telling you the angle is exactly 90°, which is more than any other value of the dot product tells you, because every other value needs an inverse cosine to interpret.
Then stop at 0°: the parallelogram has collapsed to a line, the area is 0, and the dot product is at full size, 9. The two products are reading the same geometry and reporting opposite halves of it.
| Question | Answer |
|---|---|
| 1. (3, 4, 0) · (4, −3, 0) | 12 − 12 + 0 = 0, so perpendicular. |
| 2. (1, 2, 2) · (2, 2, 1) | 2 + 4 + 2 = 8. |
| 3. The angle between them | cosθ = 8/9, so 27.3°. |
| 4. The triangle's area | v × w = (−2, 3, −2), magnitude √17 = 4.12, halved: 2.06. |
| 5. What a dot product of 0 means | B. The two are perpendicular, assuming neither is the zero vector. |
Both vectors in questions 2 to 4 have magnitude 3, which is deliberate: |v||w| = 9, so cosθ = 8/9 with no intermediate rounding, and the angle is clean. It also means the figure's bars, which are 9cosθ and 9sinθ, are exactly this question's numbers.
1 markThe product from components.
1 markBoth magnitudes.
1 markcosθ, then θ in degrees.
1 markThe halving, on a triangle.
Two of these four are final-step marks, and both are lost by stopping one line early: 0.889 instead of 27.3°, and 4.12 instead of 2.06. Worth saying plainly that an unfinished answer scores like a wrong one.
| They wrote | What happened |
|---|---|
| 24 on question 1 | Dropped the minus sign in the second vector: 12 + 12. Catch it here, because the whole answer depends on the sign. |
| 25 on question 1 | Multiplied the magnitudes, 5 × 5. Interestingly that IS |v × w| for this pair, since they are perpendicular, so it is right as an area and wrong as a dot product. |
| 9 on question 2 | |v||w| rather than v · w. They have used the geometric form with cosθ set to 1. |
| 5 on question 2 | Added the first vector's components. They have not multiplied at all. |
| 0.889 | On question 3, stopped at cosθ. The commonest single error in the sub-topic. The giveaway is that an angle cannot be less than 1 degree here. |
| 62.7° | Inverse sine instead of inverse cosine. 90 − 27.3. |
| 0.476 | Radians. The mode, not the method. |
| 4.12 on question 4 | The parallelogram, not the triangle. Out by a factor of two and entirely plausible-looking. |
| 4 on question 4 | Halved the SCALAR product. They have reached for the wrong product; area always comes from the vector one. |
| 17 | Forgot the square root inside the magnitude. |
"How do I know which product to use?" One question: what kind of answer does it want? A number that is an angle, or a test for perpendicularity, means dot. An area, or a direction perpendicular to two others, means cross. Write the answer's type down before computing anything.
"Why does the cross product need three dimensions?" Because its answer is a direction perpendicular to both inputs, and in two dimensions there is nowhere for it to point. For a two-dimensional area question, pad the vectors with a zero third component; the cross product then has only a third component and its size is the area. Demonstrate it once with (3, 4, 0) and (4, −3, 0), where the answer is (0, 0, −25) and the area is 25.
"Does the order matter?" Not for the dot product. For the cross product it reverses the answer, so w × v = −(v × w). Magnitudes are unaffected, so area questions forgive it. A question asking for a direction does not.
"Can the angle between two lines be obtuse?" The calculation can give an obtuse value, because a direction vector can be written either way round, and the convention is to report the acute angle. Tell them to check the sign of the dot product first: negative means take 180 minus the answer. It is one line and it removes a whole class of disputes.
| Stage | What to do |
|---|---|
| Demonstrate | Store v and w, then put four lines on one screen: dotP(v,w), norm(v)*norm(w), crossP(v,w), norm(crossP(v,w)). 8, 9, (−2,3,−2), 4.123. The 8 against the 9 is the useful juxtaposition, because it shows at a glance that the angle is small, and it makes the 9-as-an-answer error look like what it is. |
| Where they stick | Finding the commands at all. On the Casio they are behind OPTN → MAT/VCT and then F6 twice, which nobody discovers by accident. On the Nspire, crossP accepts 2-element vectors as well as 3-element ones and pads the answer to (0, 0, z), which is correct and surprises students who expect an error. Show both once, slowly, and have them write the key sequence in their own notes. |
| The check | Dot the cross product back against each original. Both should give 0. It is two commands, it confirms the whole computation, and it reinforces the perpendicularity meaning of zero at the same time. |
Degrees mode before any inverse cosine on this page, or 27.3 arrives as 0.476 and looks like a different kind of error. The mode matters here and at 3.10, where resolving calls cos and sin.
| Step | What |
|---|---|
| 1 | The dot product from components, on (3,4,0) and (4,−3,0). Collect the answer. Someone will say they have gone wrong. |
| 2 | Why 0 is the result: |v||w|cosθ with cosθ = 0. |
| 3 | The sign rule: positive acute, zero right, negative obtuse. Three examples, no algebra. |
| 4 | (1,2,2) and (2,2,1): 8, then 8/9, then 27.3°. Insist on the last step. |
| 5 | Figure, sweep. The two bars trading places. |
| 6 | The cross product, with the perpendicularity check done on the board. |
| 7 | Parallelogram against triangle, and the halving written as a habit. |
| 8 | The angle between two lines, using directions only, with deliberately unhelpful starting points. |
Do not say "the dot product measures how much they point the same way". It is nearly right and it hides the magnitudes. (10, 20, 20) · (2, 2, 1) = 80, ten times (1, 2, 2) · (2, 2, 1) = 8, and (10, 20, 20) points in exactly the same direction as (1, 2, 2). The angle has not changed at all; only the length has. Say that the dot product combines the two sizes and the angle together, and that cosθ is what is left once the sizes are divided out.
Do not say "cross product gives the area". It gives a vector; its magnitude gives the area of the parallelogram. Students who learn the short version write a vector where a number belongs, or forget the halving, and both are marks. The long version takes four extra words.