Clifford product of force and distance

Question

$fd = force*distance$

$fd = (f_1\mathbf{e_1} + f_2\mathbf{e_2})(d_1\mathbf{e_1} + d_2\mathbf{e_2})$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} + f_2d_1\mathbf{e_2}\mathbf{e_1}$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} - f_2d_1\mathbf{e_1}\mathbf{e_2}$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + (f_1d_2 - f_2d_1)\mathbf{e_1}\mathbf{e_2}$

$fd = Scalar + Bivector?$

The scalar is just the energy but what is the bivector?

Answer 1

Note that in physics classical mechanics context

• for dot product $f\cdot d=f_1d_1+f_2d_2$ is the definition of work
• for cross product $f\times d=(f_1d_2-f_2d_1)\mathbf{e_3}$ is the definition of torque