User:Prof McCarthy/Rotation matrix

As far as i can tell this article on the rotation matrix was created in 2004 and by 2005 had already had the signs changed to try to find a consistent definition. The current talk page goes back to 2006, and editors have cited resources on both sides about where the minus sign should be placed. I am reluctant to wade into this discussion because each side has a consistent explanation and literature to justify their location of the minus sign.

Maybe it will help to start by saying both views are correct, which sounds strange but may help focus on what are actually two distinct ways of using a rotation matrix: (i) a transformation from coordinates in one frame to coordinates in another frame, and (ii) a transformation between two sets of basis vectors defining coordinates in the same frame. The two formulations are so similar that it is probably no surprise that they result in matrices that are inverses of each other, which for rotation matrices means the transpose of each other, and all the transpose does is move the minus sign.

(i) Consider the first case, where a reference frame M is rotated counter clockwise by the angle θ relative to a reference frame F. A vector in M has the coordinate x=xi + yj measured, where i=(1,0) and j=(0,1) are the natural basis vectors along its coordinate axes, so x,=(x, y) in this reference frame. Now consider the coordinates X=(X, Y) of the same point but now measured in F. This is easily done by considering the vectors e_r=(cosθ, sinθ) and e_t=(-sinθ, cosθ) that are the images of i and j of M but now measured in the frame F, so X=xe_r+ye_r, or

${\begin{Bmatrix}X\\Y\end{Bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{Bmatrix}x\\y\end{Bmatrix}}.$

This version of the rotation matrix has the vectors e_r=(cosθ, sinθ) and e_t=(-sinθ, cosθ) as column vectors, which means the minus sign located on the upper right sine term.

(ii) Now consider the second case, where the vectors are measured in the same frame F, so i and j are the natural basis vectors along the x and y axis of this frame, and we have the unit vectors e_r=(cosθ, sinθ) and e_t=(-sinθ, cosθ) measured in F that define a pair of orthogonal unit vectors rotated relative to the basis vectors by the angle θ in the counter clockwise direction. Now, the transformation from coordinates relative to the basis vectors i and j of F to the new basis vectors e_r=(cosθ, sinθ) and e_t=(-sinθ, cosθ) for F, is easily defined by the computing i=cosθe_r-sinθe_t j=sinθe_r+cosθe_t. This means a vector X in F is transformed to a vector in the rotated basis, by the transformation,

${\begin{Bmatrix}x\\y\end{Bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{Bmatrix}X\\Y\end{Bmatrix}}$

Now the columns of the rotation matrix are the coordinates of the natural basis vectors i and j as measured in the new basis e_r and e_t.