#### You may also like A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular? ### A Knight's Journey

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition. ### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Vector Walk

##### Age 14 to 18Challenge Level

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?

NOTES AND BACKGROUND

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.