Ytukyg

2 $begingroup$ Given a vector v and a basis A, where its coordinates are (x,y), in order to find v 's coordinates in a new basis A' (i.e. x' and y') which is a rotation of A by angle θ, it is sometimes said that you must proceed as follows: a) find the coordinates in the original basis A of the unit vectors of the new basis (say e’ x and e’ y ), which happen to be (cosθ, sinθ) and (-sin θ, cos θ), respectively; b) then x' in the new basis A' = dot product between v (coordinates in A = x and y) and e’ x (also as per coordinates in A = xcosθ +ysenθ); c) whereas y' = dot product between again v in A and e’ y in A = x(-sinθ) +ycosθ. In matrix notation, the coordinates in the original basis A of the unit vectors of the new basis form the following matrix: $$left( {begin{arra...