Summetros
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I was given the following as a proof that the inertial tensor was symmetric. I won't write the tensor itself but I will write the form of it below in the proof. I am confused about the steps taken in the proof. It involves grade projections.
[tex]A \cdot (x \wedge (x \cdot B)) = \langle Ax(x \cdot B)\rangle = \langle (A \cdot x)xB\rangle = B \cdot (x \wedge (x \cdot A ))[/tex]
A and B are bivectors and x is a vector. So x dot B is a vector and the outer product of x and x dot B is a bivector. So the A dot that is a bivector dotted with another bivector, so the whole thing is grade 0 (scalar). Is this why you can put all the terms inbetween the left and right angle brackets and do what ever with them (provided order is maintained)?.
I guess my question is what are and are not okay manipulations when doing these sorts of grade projections? Why can the dot be switched from one set of terms to another as shown in the middle two expressions? Whys can you rewrite the third expression as the last expression?
[tex]A \cdot (x \wedge (x \cdot B)) = \langle Ax(x \cdot B)\rangle = \langle (A \cdot x)xB\rangle = B \cdot (x \wedge (x \cdot A ))[/tex]
A and B are bivectors and x is a vector. So x dot B is a vector and the outer product of x and x dot B is a bivector. So the A dot that is a bivector dotted with another bivector, so the whole thing is grade 0 (scalar). Is this why you can put all the terms inbetween the left and right angle brackets and do what ever with them (provided order is maintained)?.
I guess my question is what are and are not okay manipulations when doing these sorts of grade projections? Why can the dot be switched from one set of terms to another as shown in the middle two expressions? Whys can you rewrite the third expression as the last expression?