### Why do this problem?

This problem gives students useful practice in working with
vectors and matrices, in the context of operation which preserve
the structure of a crystal. Students have the opportunity to use
their geometrical and algebraic reasoning skills to answer the
questions.

### Possible approach

Start by sharing with students the coordinate definition of
the crystal structure, and ask them to make sketches to show what
the structure looks like.

Discuss what might be meant by "preserving the structure of
the crystal" and ask students to come up with some suggestions of
transformations represented by constant matrices or constant
vectors which could be used to transform the crystal while
preserving its structure.

Once students have had some opportunity to consider suitable
transformations, set them the problem of working out which of the
given matrices and vectors are suitable candidates. Challenge them
to explain geometrically as well as algebraically what happens when
each matrix is applied or each vector is added.

### Key questions

What do we mean when we say the structure of the crystal was
preserved by a transformation?

What transformations will always/sometimes/never preserve a
crystal structure?

### Possible extension

Clear as
Crystal poses challenging questions about transformations and
crystal symmetries.

### Possible support

Coordinated
Crystals gives an introduction to the idea of representing
crystal lattices using coordinate geometry.