# Vector problems in high school precalculus

I was helping my kid study for a precalculus examination and looking at her old tests from the year, and came across a question about vectors. Below is the typed up version of the question and her answer; an image from the actual test can be found at the end.

6. A vector with tail at the origin has a magnitude of $5$ and a direction angle of $\pi/3$. When trying to find the coordinates of the head, Nick says, "That's easy. It's just $r \cos \theta$ and $r \sin \theta$; we've been doing this forever now!" Olathe counters, "No, we should use the idea of components to find out how much our vector goes along each axis."

Who's correct here? Give a thorough, mathematically sound, explanation!

The student writes:

Nick is correct. As long as we have the magnitude and directional angle, and know that the tail is located at the origin, the method he proposed is suitable. (Crossed out: Olathe is correct. Even though we have the magnitude and directional $\theta$ we can't) Components are unnecessary b/c we know the tail is at the origin, we know how far out the vector goes, and we know the $\theta$ it makes with the $x$-axis.

I'm not sure how thorough this needed to be, but I agree with the kid. Or I agree that both approaches are equivalent - maybe that was the point. Would any high school math educators care to weigh in?

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I don't understand what about this question relates to math education. It just seems like a straight math question. –  Ben Crowell Jun 11 at 19:36
@BenCrowell, a fine point... but it is about how a specific question should be graded, so education-related. But perhaps to far along the "opinion only" line... –  vonbrand Jun 11 at 20:00
this is a horribly vague question. Perhaps what the teacher wanted is a statement like, "Olathe's description is vague. How would you find these components. You can't just draw the vector, because your drawing will not be exact. Since the tail is at the origin, the way to find the components is exactly what Nick described, and the head is at (r cos theta, r sin theta), both students are correct, but Nick is more specific and actually proposes a method." The teacher may have also wanted a proof of (r cos theta, r sin theta). –  MHH Jun 13 at 13:30

I feel that the student demonstrated the relevant understanding here, but my guess is that the teacher was looking for some acknowledgement of the validity of Olathe's point of view and/or the equivalence of the two points of view.

This type of question is tricky to grade for precisely this reason. It seems to me the question was designed to elicit discussion of the relationship between the two points of view, and points were deducted because the answer didn't do this. This is hazy judgement territory though because (IMHO) it was reasonable for the student to take the language of the question at face value and opine about whose point of view was better, especially since they referenced the key fact that the tail is at the origin which makes Nick's method valid.

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Thanks for that. For those who don't have difficulty reading the graphic, you can see that scribbled out was the opposing answer as well. Maybe my kid was hedging her bets. –  Essellgee Jun 11 at 16:51

The analysis of Nick's approach is ok, although the first part is just a shortened repetition of the information given in the task.

The question indeed asks for an analysis of both approaches. A comparison of both approaches isn't necessary, although an equivalence argument can be used instead of the second analysis.

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The question indeed asks "who's correct here?", but I would agree high school students should always be primed to compare and contrast in any written answer. –  Essellgee Jun 11 at 18:13
High school students should always be primed to do the task defined by the operator used. Here, the first is a question without operator. But the operator should be Judge!. The second has an operator: Explain!. Nowhere is an operator asking for a comparison. So why should a high school student compare? –  Toscho Jun 11 at 18:20
Actually, the first thing Olathe said before presenting his method was "no...", so without wading too deep into semantic silliness, HE cannot be correct (although his approach might be) if the methods are equivalent. –  Essellgee Jun 11 at 18:24
The "No!" concerns his heuristic, not his mathematical methods. –  Toscho Jun 11 at 18:30