You may also like

problem icon

Flexi Quads

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?

problem icon

Flexi Quad Tan

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

problem icon

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.

Vector Walk

Age 14 to 18 Challenge 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?


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.