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Cross Product. Is there any other basis than the Right Hand Rule when determining the direction of the Cross Product?

Is there a Mathematical or Scientific, or any explanation/proof that points the direction of the Cross Product?

9 x 23 sin(30) = 108 1/2 = 54... Ok, So I got c = 54... but doesn't tell what direction, just basing on the Right Hand Rule.

m0Onfang
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  • It's a convention. More precisely, the cross product depends on orientation in the given vector space. The orientation can be thought as a choice of a basis, and if you map the basis to another basis, the determinant of this linear map is positive iff the bases have the same orientation. We have "right hand rule" because of the choise $\boldsymbol{i},\boldsymbol{j},\boldsymbol{k}$ basis. – Canis Lupus Oct 08 '16 at 12:52

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One way to define the cross product is to declare that $a\times b$ is the unique vector such that for every $c$: $$ (a\times b) \cdot c = \det [a,b,c]$$ This defines the components, whence the direction, of $a \times b$ uniquely, e.g. $(a\times b)_z=a_xb_y-a_yb_x$, etc... But of course this leaves the definition of direction in the hands of $\det(\cdot)$.

H. H. Rugh
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