Question

Consider a vector A in two dimensions with components (Ax, Ay). Find the components (A'x, A'y) in a new coordinate system (x', y') which is rotated by an angle theta counterclockwise with respect to (x, y). From this, construct a 2 x 2 matrix R(theta) such that (A'x, A'y)=R(theta)(Ax, Ay)

Answer 1

Excuse my lack of symbols and proper equation calligraphy

Answer 2

I'll try to forgive it, this time. So what do we do here, I think the best way is to draw it out and label your new vector in terms of the old using sine and cosine. Give it a shot, I'm gonna brb and pour another cup of coffee.

Answer 3

And from here I'm lost because I'm bad with rotations

Answer 4

I know it's just a matter of putting the right trig components in the matrix but I always trip over myself and second guess

Answer 5

Ah ok, so first off the vector itself is not moving, it's stuck at one place in space. What's happening is we're using two separate coordinate systems to represent the same vector, which is kinda weird. I'll help label everything to hopefully make things clearer first, then I'll show how to do part of it and let you try the next.



So now this thing has coordinates in both coordinate systems, the regular and the rotated frame