Note on Matrix of Rotation
Total Page：16
File Type：pdf, Size：1020Kb
Note on matrix of rotation Matrix of rotation in the x; y-plane We rotate the system of coordinates x; y by the angle α into the system of coordinaes x0; y0. Then the versors (unit vectors) ^i and ^j are rotated by the angle α into new versors i^0 and j^0 (see Figure 1). We express the new versor i^0 in the coordinates x; y as i^0 = cos α^i + sin α ^j (1) because the value of the projection of the versor i^0 onto the Ox axis is cos α and the value of the projection of the versor i^0 onto the Oy axis is sin α. Similarly, we have the following relation for the versor j^0 j^0 = − sin α^i + cos α ^j (2) because the value of the projection of the versor j^0 onto the Ox axis is − sin α and the value of the projection of the versor j^0 onto the Oy axis is cos α. Any vector ~v can be expressed in coordinates Oxy ~v = vx ^i + vy ^j (3) or in coordinates Ox0y0 as 0 0 ~v = vx0 i^ + vy0 j^ (4) We can rewrite equation (4) inserting the expressions for i^0 from the equa- tion (1) and for j^0 from the equation (2) ~v = vx0 (cos α^i + sin α ^j) + vy0 (− sin α^i + cos α ^j) (5) = (vx0 cos α − vy0 sin α)^i + (vx0 sin α + vy0 cos α)^j 1 The equation (5) expresses the vector ~v with primed coordinates vx0 and vy0 in original unrotated coordinate system Oxy. Comparing equation (5) with equation (3) we see that vx0 cos α − vy0 sin α = vx (6) vx0 sin α + vy0 cos α = vy We may solve the system of linear equations (6) to calculate vx0 and vx0 , i.e. the coordinates of the vector ~v in the rotated coordinate system Ox0y0. The determinants D, Dvx0 , Dvy0 are cos α − sin α 2 2 D = = cos α + sin α = 1 (7) sin α cos α v − sin α D = x = v cos α + v sin α (8) vx0 x y vy cos α cos α v D = x = −v sin α + v cos α (9) vy0 x y sin α vy 0 0 and therefore using the formulas vx = Dvx0 =D and vy = Dvy0 =D we obtain vx0 = vx cos α + vy sin α (10) vy0 = −vx sin α + vy cos α Applying the matrix notation we have " # " #" # v 0 cos α sin α v x = x (11) vy0 − sin α cos α vy The above matrix equation (11) gives the coordinates of original vector ~v in rotated system Ox0y0. So it is as if the vector in the primed system Ox0y0 is 0 0 rotated by the angle −α. Then the coordinates vx and vy of the vector as rotated by angle α will be " # " #" # v0 cos α − sin α v x = x (12) 0 vy sin α cos α vy if the vector had original coordinates vx and vy because we had to change the sign by angle α to the opposite one. The matrix in the equation (12) is the rotation matrix of a point of coordinates vx and vy by the angle α. 2 Pawel Jan Piskorz ([email protected]) 3 Figure 1: We rotate the system of coordinates x; y by the angle α into the system of coordinaes x0; y0. Then the versors (unit vectors) ^i and ^j are rotated by the angle α into new versors i^0 and j^0. 4.