Is this a group?
For vectors ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) in R 3 , define the cross product × as such:
( x 1 , y 1 , z 1 ) × ( x 2 , y 2 , z 2 ) = ( y 1 z 2 − z 1 y 2 , z 1 x 2 − x 1 z 2 , x 1 y 2 − y 1 x 2 )
Does R 3 form a group under the cross product?
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1 solution
A group G with some binary operator ∗ is defined by the 3 group axioms:
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∀ a , b , c ∈ G ( ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) ) This means that the operator is associative.
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∃ e ∈ G ∀ a ∈ G ( a ∗ e = e ∗ a = a ) This means that there exists an identity element.
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∀ a ∈ G ∃ b ∈ G ( a ∗ b = b ∗ a = e ) This means that there exists an inverse element.
Another way to show that ( R 3 , × ) is not a group is to show that product of any vector with the zero vector ( 0 , 0 , 0 ) is the zero vector, so no inverse element exists for it.
The cross product of two vectors is orthogonal to the original two vectors, so an identity element cannot exist.