Note on matrix of

Matrix of rotation in the x, y- We rotate the system of coordinates x, y by the α into the system of coordinaes x0, y0. Then the versors ( vectors) ˆi and ˆj are rotated by the angle α into new versors iˆ0 and jˆ0 (see Figure 1). We express the new versor iˆ0 in the coordinates x, y as

iˆ0 = cos αˆi + sin α ˆj (1) because the value of the projection of the versor iˆ0 onto the Ox axis is cos α and the value of the projection of the versor iˆ0 onto the Oy axis is sin α. Similarly, we have the following relation for the versor jˆ0

jˆ0 = − sin αˆi + cos α ˆj (2) because the value of the projection of the versor jˆ0 onto the Ox axis is − sin α and the value of the projection of the versor jˆ0 onto the Oy axis is cos α. Any vector ~v can be expressed in coordinates Oxy

~v = vx ˆi + vy ˆj (3) or in coordinates Ox0y0 as

0 0 ~v = vx0 iˆ + vy0 jˆ (4)

We can rewrite equation (4) inserting the expressions for iˆ0 from the equa- tion (1) and for jˆ0 from the equation (2)

~v = vx0 (cos αˆi + sin α ˆj) + vy0 (− sin αˆi + cos α ˆj) (5)

= (vx0 cos α − vy0 sin α)ˆi + (vx0 sin α + vy0 cos α)ˆj

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) ˆi and ˆj are rotated by the angle α into new versors iˆ0 and jˆ0.

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