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.

Limiting Probabilities

Given probabilities of taking paths in a graph from each node, use matrix multiplication to find the probability of going from one vertex to another in 2 stages, or 3, or 4 or even 100.

Transformations for 10

Stage: 5 Challenge Level:

The equations for a line and plane in vector form may be useful.

Line: ${\bf r}={\bf a} + \lambda{\bf b}$
Plane: ${\bf r}={\bf a} + \lambda{\bf b}+ \mu{\bf c}$

It may also be useful to recall that matrix multiplication is distributive:
${\bf M}({\bf a} + {\bf b}) = {\bf Ma} + {\bf Mb}$

This problem uses concepts met in the later Further Pure Maths A level modules.