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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?

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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.

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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: Challenge Level:2 Challenge Level:2

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.