The Student Room Group

Transformations using matrices

Find the 2×22\times2 matrix AA corresponding to to a rotation by 4545^{\circ}anticlockwise about the origin. How would I mathematically work this out rather than just stating it?
Original post by miketree
Find the 2×22\times2 matrix AA corresponding to to a rotation by 4545^{\circ}anticlockwise about the origin. How would I mathematically work this out rather than just stating it?


Why do you want to work it out?
Original post by miketree
Find the 2×22\times2 matrix AA corresponding to to a rotation by 4545^{\circ}anticlockwise about the origin. How would I mathematically work this out rather than just stating it?



I don't know if this will help, but I don't memorise the matrix and am too lazy to look it up, so I re-derive it each time I need it.

If we know what the matrix does to the vectors (1,0) and (0,1), we've got the whole thing. (In your notation, those two vectors ought to be written in columns, of course.)

So I draw the x-y axes, and then draw (1,0) rotated by theta. It obviously goes to: (cos theta, sin theta)

Similarly, drawing where (0, 1) goes on the axes, I see it goes to: (-cos theta, sin theta) So what we want is a matrix such that:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} \cos ( \theta) \\ \sin ( \theta ) \end{pmatrix}

And this forces a = cos theta and c = sin theta. Doing the same thing for (0, 1) yields the other two values.


Posted from TSR Mobile
(edited 9 years ago)
Reply 3
Original post by majmuh24
I don't know if this will help, but I don't memorise the matrix and am too lazy to look it up, so I re-derive it each time I need it.


Perfect, thank you!

Quick Reply