Proof for cross product a x b= -b x a

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SUMMARY

The discussion centers on proving the vector identity \( \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}) \). Participants emphasize that the proof can be achieved by calculating the cross product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \) in their component forms and demonstrating that the resulting coordinates are the negative of those obtained from \( \mathbf{b} \times \mathbf{a} \). The consensus is that showing the equality of the two cross products component-wise suffices as a valid proof.

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senorclean
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This problem involves vectors and it's a fairly basic proof but I can't seem to wrap my head around it.

I tried to just separate the vectors into their components and cross them but then I just get another set of coordinates that doesn't seem factorable. I really can't see any way to manipulate it.

I thought about just doing the cross product of -(b x a) and showing that the coordinates would be the negative version of a x b but I'm not sure if that's even proving anything. I hate proofs. Any guidance would be greatly appreciated.
 
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Yes,doing the cross product of a x b and showing it is equal (component wise) to -b x a is enough.
 
ok then. gracias
 

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