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Is there a simple proof that the cross product (defined as the usual determinant) always obeys the right hand rule?

user85798
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1 Answers1

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By the linearity and anticommutativity, it suffices to prove that $i\times j=k$. $$\left|\begin{array}{ccc}i&j&k\\1&0&0\\0&1&0\end{array}\right|=k$$

A similar computation proves that $j\times k=i$ and $k\times i=j$.

vadim123
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  • I see that it works for the unit vectors i,j,k- but I don't understand how linearity and anticommutativity imply it must be true for all vectors. – user85798 Feb 16 '14 at 00:49
  • The only reason you need the RHR is to determine $\pm$; the direction is specified as normal to the $u-v$ plane, and the length is $|u||v|\sin\theta$. – vadim123 Feb 16 '14 at 02:46