Show that the edges AD and BC of a tetrahedron ABCD are mutually perpendicular when: AB²+CD² = AC²+BD².

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The article provides a summary of the elementary ideas about vectors usually met in school mathematics, describes what vectors are and how to add, subtract and multiply them by scalars and indicates. . . .

Can you combine vectors to get from one point to another?

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

An account of multiplication of vectors, both scalar products and vector products.

Find out how the quaternion function G(v) = qvq^-1 gives a simple algebraic method for working with rotations in 3-space.

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?

Take any parallelogram and draw squares on the sides of the parallelogram. What can you prove about the quadrilateral formed by joining the centres of these squares?

A short introduction to complex numbers written primarily for students aged 14 to 19.

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Can you find the area of a parallelogram defined by two vectors?

Go on a vector walk and determine which points on the walk are closest to the origin.

Explore the lattice and vector structure of this crystal.

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

Analyse these repeating patterns. Decide on the conditions for a periodic pattern to occur and when the pattern extends to infinity.

Can you arrange a set of charged particles so that none of them start to move when released from rest?

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.