How are exercises in vector geometry created?

Question

I'm fairly new to teaching and am currently teaching high school students about vector geometry. I am currently making a test for them for next week and it is mostly about the scalar product and the vector product. For this, I went through my old exams and exercises (which are a lot) and chose the ones I thought were important to solve.

Now there is one exercise that I think would be a good exercise for them, however it has really really ugly numbers as a result whereas all other exercises in the exam have very nice numbers. It reads as follows:

The point $M = (6,1,6)$ is midpoint of the regular octahedron $ABCDEF$. Further, one of the edges is part of the line $g$ consisting of the points $P = (0,-6,6)$ and $Q = (9,0,3)$. Find the volume of the octahedron without calculating any futher points. Then, find the coordinates of the six vertices $A,B, C, D, E, F$.

What software would be suited to create such exercises? Basically, what I need is an octahedron with a "nice" volume, its vertices at nice points in $\mathbb{R}^3$ and one of its edges running through nice points. By nice, I mean points with coordinates in $\mathbb{Z}$ or maybe very simple fractions.

Answer 1

I'm not sure how software would be helpful here. To make a problem of this form, you need to:

1. Choose any center point $M$.

2. Choose two orthogonal vectors $\textbf{v}$ and $\textbf{w}$ of the same length.

3. Compute $M + \textbf{v}$ and $M+\textbf{w}$.

4. Find two other points on the line through $M+\textbf{v}$ and $M+\textbf{w}$.

The volume will end up being $\dfrac{4}{3}\|\textbf{v}\|^3$. If you want to avoid radicals during the intermediate calculations, the only important consideration is that $\textbf{v}$ and $\textbf{w}$ have integer lengths.

Here are some orthogonal triples of integer-length vectors of equal lengths that you can use to pick $\textbf{v}$ and $\textbf{w}$.

• $(3,4,0)$, $(4,-3,0)$, and $(0,0,5)$. More generally, $(a,b,0)$, $(-b,a,0)$, and $(0,0,c)$ for any Pythagorean triple $(a,b,c)$.

• $(1,2,2)$, $(2,1,-2)$, and $(-2,2,-1)$

• $(2,3,6)$, $(6,2,-3)$, and $(3,-6,2)$

• $(4,4,7)$, $(1,-8,4)$, and $(8,-1,-4)$

If you look for them, these triples can be found in many vector geometry problems.

Answer 2

Trial-and-Error solution

Model this situation with any Dynamic Geometry Software, e.g. GeoGebra, using parameters. Change the parameters, e.g. using slow animation, watching for "nice" solutions.