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

In summary, the problem involves vectors and their components, and the goal is to prove the equality of -(b x a) and a x b. The conversation discusses different approaches, including separating the vectors and taking the cross product, but ultimately concludes that using the cross product of a x b and showing its equality with -b x a is sufficient to prove the statement.
  • #1
senorclean
2
0
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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  • #2
Yes,doing the cross product of a x b and showing it is equal (component wise) to -b x a is enough.
 
  • #3
ok then. gracias
 

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

What is the proof for cross product a x b = -b x a?

The proof for the cross product a x b = -b x a can be shown using the properties of the cross product and the properties of vectors in three-dimensional space.

What are the properties of the cross product?

The properties of the cross product include the distributive property, the anti-commutative property, and the magnitude property. These properties help to define and understand how the cross product behaves when multiplied with other vectors.

How do you prove the distributive property of the cross product?

The distributive property of the cross product can be proven using the definition of the cross product and the properties of vectors. By expanding the cross products and simplifying, it can be shown that a x (b + c) = a x b + a x c, and (b + c) x a = b x a + c x a.

What is the anti-commutative property of the cross product?

The anti-commutative property of the cross product states that the order of the vectors in the cross product does not affect the magnitude or direction of the resulting vector. In other words, a x b = -b x a.

How does the magnitude property of the cross product relate to its proof?

The magnitude property of the cross product states that the magnitude of the resulting vector is equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them. This property is used in the proof to show that the magnitude of a x b is equal to the magnitude of -b x a, further supporting the anti-commutative property.

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