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?