Prove that the torque acting due to a force F in the xy - plane is ?x = xFy - yFx.

we know that the torque is given as:

? = r X F....r=perpendicular distance of the applied force from the fixed point,
? =torque,
F=force

here

in the xy plane

r = ix + jy

F = iF_x + jF_y

so, we have

? = r X F = ( ix + jy) X (iF_x + jF_y)

by solving further using matrix method, we get

? = ix X (iF_x + jF_y) - jy X (iF_x + jF_y)

so, finally

? = xF_y - yF_x