# Matrix multiplication/Rotations

An engineer came to me with the following problem

## Homework Statement

Suppose
$y = A x$
where $x,y \in R^2$
and $A = \left( \begin{array}{cc} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{array} \right)$.
Show that y is an anticlockwise rotation of x about the origin.

## Homework Equations

None.
Maybe definition of SO(2,R).

## The Attempt at a Solution

$\square$

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I kind of don't understand the question.
How can you prove a definition?
Is this question not asking something like, "prove that average speed equals distance over time"?
I guess they want the student to draw a load of triangles? Or perhaps express x and y in terms of polar coordinates to make it more obvious that it's a rotation? Or maybe to show that it can be written as a product of two reflections? Or show that |x| = |y|...but that doesn't make any comments about the angle.
Any suggestions?
Thanks

i think what is being asked of you is to forget about SO(2) and everything. Just draw two point vectors in a 2-d plane separated by an angular distance of \theta.

Now, by just applying your knowledge of geometry, show that the new coordinates of y in terms of the old coordinates x, is the exact same equations as written above in matrix form.

HallsofIvy