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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?

Rots and Refs

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Reflect Again

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

Transformations for 10

Age 16 to 18
Challenge Level

Why do this problem?

This problem gives a good opportunity to think about the action of matrices on vectors in geometrical as well as algebraic terms, allowing students to develop ideas about what multiplication of a vector by a matrix actually means.

Possible approach

One approach that works well is to divide the class into groups and give each group some of the questions to work on. A possible grouping is questions 1-3, 4-7, and 8-10, with 8-10 being the most challenging.

Ask each group to first read through each question and decide whether they have any intuitive feel for what the right answer might be. Then they should use algebra and/or geometrical arguments to justify their answers. In cases where the answer depends on various factors, students should clearly explain what these factors are. Explain that at the end of the session they will have to justify their answers to the rest of the class, so they should prepare a presentation to explain their findings.

Some students may need reminding about the form of the vector equations of a line and a plane.

At the end, allow plenty of time for students to present their answers to the questions they were given, and encourage the rest of the class to be critical, asking questions and challenging anything that doesn't make sense to them.

Key questions

Can you give an algebraic example to justify your answer to the question?
How does a geometrical interpretation of the situation help?

Possible extension

Questions 8-10 are a little more challenging than the first few questions in the problem.
Matrix Meaning extends students' understanding about the effects of matrices which reflect or rotate in two or three dimensions.

Possible support

Square Pair gives a good starting point for considering how matrices transform vectors in two dimensions.