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Unformatted text preview: A determinant is a mathematical operation that completely antisymmetrizes a
square matrix. On a 2 µ 2 matrix we have: ab
On a higher order matrix we write it as an alternating sum of determinants of lower order. We will consider only the case of the cross product. ```
` A y Az
` A x Az
` Ax Ay
A ä B = A x A y Az = x
B y Bz
B x Bz
B x B y Bz
= x I A y Bz - Az B y M - y H A x Bz - Az B x L + z I A x B y - A y B x M
Note that there is not a discrepancy between the differing signs of the middle term for the y-component, since the factor multiplying y has terms
in reverse order. Angular Momentum of a Particle and Torque
We previously defined the torque as ”
t = r ä F.
We can similarly define the angular momentum of a particle as ”
L = r ä p.
Both of these expressions are relative to an origin; r is the position vector. It is from the origin to the position to where the force is applied in
the case of torque. It is from the origin to the position of the particle for the angular momentum.
The net torque on a particle (a point) is the torque due to the net force. If the particle is at position r then the net torque is tnet = r ä F net . It is
straightforward to verify that the cross product satisfies the usually product rule for differ...
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