Creative problems in 2D vector geometry

Question

What are some "interesting" and creative problems or exercises on specifically 2-dimensional vector geometry that a high school student might find compelling to solve?

The class' current knowledge consists of basic vector arithmetic, parametric vector equations of lines, the scalar dot product and pretty much everything that can be done with it algebraically and geometrically. No matrices are known at this stage.

I can think of many "mathematically glamorous" problems in 3D vector geometry, but not in 2D. It seems to me that any 2D vector problem (at high school level) instantly reduces to either some standard application of the scalar product or instantly translates into a linear system of equations, and that there is little room to pose something with more depth. Any ideas would be appreciated.

Answer 1

A modestly non-trivial investigation that yields well to vector algebra is the Euler line. That is, showing that the centroid of any triangle lies on the line segment connecting its circumcenter to its orthocenter and divides that line segment in the ratio 2:1.

Answer 2

I just had some students implement reflection of a ray in a mirrored line segment, which they then used to bounce a light ray around inside a polygon. Here's a crude snapshot:

     

This project also involved programming, but even hand calculations inside an equilateral triangle are of interest. As a bonus, this touches on an open problem:

Does every triangle have a periodic billiard path?

See this Diana Davis YouTube video for what's known on this unsolved problem.

Answer 3

Distance from a line segment to a point? With the right graphing system (maybe Desmos, but it's a bit awkward there) you can use this to draw thick line segments.

There's also barycentric coordinates in a triangle

Answer 4

The proof that the angle between two bodies which collide elastically is $90^o$ is an interesting two dimensional vector algebra problem.

Answer 5

Here is a problem that I remember running into on an Olympiad which is not too bad with vectors and very hard without.

Let $ABCDE$ be a convex pentagon. Let $V$, $W$, $X$, $Y$ and $Z$, respectively, be the midpoints of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$ and $\overline{EA}$ respectively. Suppose that there is a point $P$ in the intersection $\overline{DV} \cap \overline{EW} \cap \overline{AX} \cap \overline{BY}$. Show that the point $P$ also lies on the line segment $\overline{CZ}$.

Solution:

Translate $P$ to be at the origin. The condition that $P$ lies on $\overline{DV}$ says that $D$ and $V = \tfrac{A+B}{2}$ are parallel vectors, so $\det(D, \tfrac{A+B}{2})=0$. We deduce that $\det(D,A)+ \det(D,B)=0$ or, in other words, $\det(D,A) = \det(B,D)$. Keeping going in this manner, $\det(A,C) = \det(D,A) = \det(B,D) = \det(E,B) = \det(A,E)$. Then $\det(C, \tfrac{A+E}{2})=0$, showing that $C$ and $\tfrac{A+E}{2}= Z$ also lie on a line through the origin.

Answer 6

It might be good to prove various trigonometric identities using vector operations.

One could use vector methods to derive the laws of reflection and refraction from the Huygens construction.

Along the lines of @JosephORourke's reflection problem, one could derive vector-forms of the laws of reflection and refraction (so that one can handle interfaces that are not necessarily along the standard xy-axes), then use them to image "objects" in an optical system (for example, a thick lens).

Given two point charges of opposite sign and unequal magnitude, use vector methods to prove that the zero-equipotential is a circle [taking the potential at infinity to be zero]. (Apollonius circle.)

Answer 7

From a previous answer on SE Math, the following is offered -

Although the math behind the conformal transformation may seem obscure, the mathematical process is rather direct. Necessary elements in the transformation are a transform circle ζ of radius a and nondescript vector Z extending from the origin to various positions on the perimeter of ζ. The length of vector Z from the origin to the circumference of ζ changes through the continuum of increasing positive angles θ being swept out by Z. For each angular position taken by Z, vector $b^2$/Z is poised in the interior of ζ at corresponding angle -θ. The airfoil contour in z(x,y) space is defined by the vector sum Z + $b^2$/Z.