Vector walk
Problem
Suppose that I am given a large supply of basic vectors $b_1=\pmatrix{2\cr 1}$ and $b_2=\pmatrix{0\cr 1}$.
Starting at the origin, I take a 2-dimensional 'vector walk' where each step is either a $b_1$ vector or a $b_2$ vector, either forwards or backwards.
Investigate the possible coordinates for the end destinations of my walk.
Can you find any other pairs of basic vectors which yield exactly the same set of destinations?
Can you find any pairs of basic vectors which yield none of these destinations?
Can you find any pairs of basic vectors which allow you to visit all integer coordinates?
In more formal mathematics, the points visited by such a vector walk would be called a lattice and the two basic vectors would be called the generators. Lattices which repeat themselves are structurally interesting; the symmetry properties of such lattices are important in both pure mathematics and its applications to, for example, the properties of crystals.
Getting Started
What is the simplest horizontal step you can make by combining $b_1$ and $b_2$ steps? What about vertically?
This problem builds on GCSE level vector work, and provides a foundation for concepts met in the later Core A Level modules.
Student Solutions
Teachers' Resources
Why do this problem?
This problem encourages students to think about vectors as representing a movement from one point to another. The need for coordinate representation of points will emerge automatically and the problem naturally requires an interplay between geometry and algebra.
Possible approach
Key questions
What do the points you can reach with $b_1$ and $b_2$ have in common?
Possible support
Work systematically combining $b_1$ steps with $b_2$ steps, recording the points visited.
Possible extension
Polygon Walk explores vector walks which form polygons around the origin.