Mathematical Physics Problem Consider the transformation fro

Mathematical Physics Problem:

Consider the transformation from a two-dimensional Cartesian 12-system to a Cartesian 1 \'2\'-system, which includes both an inversion and a rotation as shown in the drawing below. Express the vector V = 3e_1 + 2e_2 in the 1\'2\'-system. Express T = ij e_i e_j in the 1 \'2\'-system. Do not forget to sum over the repeated i and j subscripts.

Solution

If a point is denoted by (x, y) then let us say that it is denoted by (x\', y\') in the new system,

then x\'= xcos(theta) + ysin(theta)
and y\' = -xsin(theta) + ycos(theta)

lets assume that axis 1 is denoted by x and axis 2 is denoted by y and so on for 1\' and 2\'
therefore V = 3e1+2e2 which is denoted by the point (3,2) in the new system it will be denoted by
x\' = 3cos(theta)+2sin(theta)
y\'= -3sin(theta) + 2cos(theta)
SO new vector V\' = (3cos(theta)+2sin(theta))e_1\'+(-3sin(theta) + 2cos(theta))e_2\'
for the b part similarly the equation of plane vector T is found by multiplying the new cordinates SO the new vector T\' = i\'.j\'e_1\'.e_2\' therefore
T\' = (icos(theta) + jsin(theta))(-isin(theta) + jcos(theta))e_1\'.e_2\'