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Derpy gets rotated by a rotation matrix

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@MCBETA

generic facts about vectors, matrices, coordinate systems

A 2×2 matrix like:
⎡ a  b ⎤  
⎣ c  d ⎦
describes how some point with coordinates [x, y] is transformed into the new point [x’, y’]:
x' = a⋅x + b⋅y  
y' = c⋅x + d⋅y
This transformation can also be understood as a combination of two column vectors, [a, c] and [b, d], scaled by the coefficients x and y, respectively:
⎡x'⎤   ⎡a⎤     ⎡b⎤  
⎢  ⎥ = ⎢ ⎥⋅x + ⎢ ⎥⋅y  
⎣y'⎦   ⎣c⎦     ⎣d⎦
This also can be seen as calculating a vector with coordinates x and y, but within a coordinate system defined by the vectors [a, c] and [b, d], instead of the usual coordinate system defined by [1, 0] and [0, 1].

Derpy rotation matrix

Now, let’s talk about this specific matrix with column vectors [cos(φ), sin(φ)] and [-sin(φ), cos(φ)]. The first is the vector [1, 0] rotated by an angle φ, and the second is the vector [0, 1] rotated by the same angle. These rotated vectors come from the definitions of sine and cosine, either as coordinates on a unit circle or as ratios in a right triangle.
When you combine these rotated vectors with coefficients x and y:
⎡x'⎤   ⎡cos(φ)⎤     ⎡-sin(φ)⎤  
⎢  ⎥ = ⎢      ⎥⋅x + ⎢       ⎥⋅y  
⎣y'⎦   ⎣sin(φ)⎦     ⎣ cos(φ)⎦
you get a vector [x, y] in a coordinate system that is a rotated version of the standard coordinate system defined by the vectors [1, 0] and [0, 1]. Because it’s defined by almost the same pair of vectors, but both rotated by φ.
——

It’s not me rotating around you. It’s you who is rotating inside me.

Alternative way, which gives exactly the same expressions: let’s define the new coordinate system defined by vector [x, y] and its orthogonal friend [-y, x]. Let’s calculate coordinates in default coordinate system of vector with coordinates [cos(φ), sin(φ)] in this new coordinate system:
[x,y]⋅cos(φ) + [-y,x]⋅sin(φ) = 
[x⋅cos(φ)-y⋅sin(φ), y⋅cos(φ)+x⋅sin(φ)] =
[x', y']

x' = x⋅cos(φ)-y⋅sin(φ)
y' = x⋅sin(φ)+y⋅cos(φ)
This is actually the rotation of vector with coordinates [1, 0] in this new coordinate system [x,y];[-y,x]. Which is also the rotation of the vector with coordinates [x,y] in the original coordinate system. Because vector with coordinates [1,0] in the new coordinate system is vector with coordinates [x,y] in the original coordinate system.
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