The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

## Vectors and matrices

This teacher package brings together all *Plus* articles on vectors and matrices.

We've grouped the articles into three categories:

- Vectors and matrices in geometry — whether it's camouflage, computer movies, or simply getting from A to B, vectors are a great way of describing the space around us and even within us;
- Vectors and matrices in physics and biology — forces have a magnitude and a direction... .
- Matrices in game theory and computer science — Need to control a virtual world, find the best strategy for world domination, or create artificial intelligence? Matrices are the best way to store those vital bits of information.

We also present a list of relevant problems from our sister site NRICH.

### Vectors and matrices in geometry

It's all in the detail — The computer animation used in movies and games is now so lifelike, it is very hard to believe that you are actually watching a surface built from simple shapes of triangles. This article how mathematics helps bring these models to life.

Finding your way home without knowing where you are — Foraging ants have a hard life, embarking on long and arduous trips several times a day, until they drop dead from exhaustion. The trips are not just long, they also follow complex zig-zag paths. So how do ants manage to find their way back home? And how do they manage to do so along a straight line? Their secret lies in a little vector geometry.

Maths goes to the movies — How do you make a virtual being as life-like as Gollum in the Lord of the Rings? This article explores the maths behind computer-generated movies and games. Vector geometry has a big part to play.

They never saw it coming — Most of us have heard of "stealth" — a technology used by the military to disguise craft from enemy radar. But nature's stealth fighters are not so well known — creatures that use motion camouflaging to approach their prey undetected. This article looks at the vector mathematics behind the phenomenon.

Time and motion — Whatever is so wonderful about point B that makes all the people at point A want to get there? This article uses vectors to see how to get to places.

Getting into the picture — Using the mathematics of perspective, researchers are now able to produce three-dimensional reconstructions of the scenes depicted in famous paintings. With all those sightlines and projections, vectors and matrices are an important tool.

Relating relativity — How do writers manage to commit the multi-dimensional world of their imagination to a two-dimensional page? This article explores how the vector geometry of projection can be used to understand language.

Solving Symmetry — Mathematicians pin down symmetry, using matrices to help them.

Secrets from a bathroom floor — Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. But while it sometimes seems free from worldly limitations, tiling is a very precise art, where not much can be left to chance. We can push and turn and wiggle, but if the maths is not right, it isn't going to tile.

Face to face — How would it feel to look in a mirror and see not your own reflection but instead how you would look as the opposite sex? Scientists use mathematical wizardry involving high-dimensional vectors to produce gender reversed images of faces.

### Vectors and matrices in physics and biology

Finding your way home without knowing where you are — Ants have tiny brains and appalling vision, yet they always manage to find their way home. Their secret lies in vector maths.

Understanding turbulence — The study of turbulence is used to understand a range of phenomena from the simple squirting of a jet of water to the activity of the Sun. The forces at work are described by vectors.

Reconstructing the tree of life — At the heart of Darwin's theory of evolution lies a beautifully simple mathematical object: the evolutionary tree. In this article we look at how maths is used to reconstruct and understand it.

The maths of climate change: the melting Arctic — The Arctic ice cap is melting fast and the consequences are grim. Mathematical modelling is key to predicting how much longer the ice will be around and assessing the impact of an ice-free Arctic on the rest of the planet.

If you can't bend it, model it! — This article explores the aerodynamics of footballs and helps you perfect your free kick.

How to make a perfect plane — Two lines in a plane always intersect in a single point ... unless the lines are parallel. This annoying exception is constantly inserting itself into otherwise simple mathematical statements. Burkard Polster and Marty Ross explain how to get around the problem.

### Matrices in game theory and computer science

The amazing librarian— Google's *Page Rank* algorithm relies heavily on linear algebra - it's as impressive as it is elegant.

Matrix: Simulating the world part I and Matrix: Simulating the world part II— These two articles give hands-on guides to creating simple computer models of anything from flocks of birds to forest fires. At the heart of it all sits one big omniscient matrix.

Matrix: Simulating the world Part II: cellular automata — Lewis Dartnell turns the universe into a matrix to model traffic, forest fires and sprawling cities.

Blast it like Beckham — What tactics should a soccer player use when taking a penalty kick? And what can the goalkeeper do to foil his plans? This article uses game theory to find the answers.

What computers can't do — This article looks at the life and work of wartime code-breaker Alan Turing. Find out what types of numbers we can't count and why there are limits on what can be achieved with Turing machines.

Game theory and the Cuban missile crisis — This article uses the Cuban missile crisis to illustrate the theory of moves, which is not just an abstract mathematical model but one that mirrors the real-life choices, and underlying thinking, of flesh-and-blood decision makers.

### Try it yourself with our sister site NRICH

Here is a selection of relevant problems and articles from our sister site NRICH.

Vectors - what are they? — This 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.

Multiplication of vectors — This article explores scalar and vector products.

Earth routes — Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

The use of maths in computer games — An account of how mathematics is used in computer games including geometry, vectors, transformations, 3D graphics, graph theory and simulations.

Quaternions and rotations — Find out how quaternions give a way of working with rotations in three-dimensional space.

Quaternions and reflections — See how four-dimensional quaternions involve vectors in three-dimensional space and how they give a simple algebraic method of working with reflections in planes in three dimensions.

Thasan's flight — An aircraft flies on a bearing of 070 degrees at 350 km/hour with wind blowing at 40 km/hour from 340 degrees. Find the actual speed and bearing of the aircraft.

The matrix — Investigate the transformations of the plane given by some 2 by 2 matrices.

Reflect again — Investigate the matrix which gives a reflection of the plane in a given line.

Rots and refs — Use a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.