K^3/Cross product/Properties/Fact

The cross product on ${}K^{3}$ fulfills the following properties (where ${}x,y,z\in K^{3}$ and ${}a,b\in K$).

1. We have

   ${}x\times y=-(y\times x)\,.$

2. We have

   ${}(ax+by)\times z=a(x\times z)+b(y\times z)\,$

   and

   ${}z\times (ax+by)=a(z\times x)+b(z\times y)\,.$

3. We have

   ${}x\times y=0\,$

   if and only if ${}x$ and ${}y$ are linearly dependent.

4. We have

   ${}x\times (y\times z)+y\times (z\times x)+z\times (x\times y)=0\,.$

5. We have

   ${}\left\langle x\times y,z\right\rangle =\det(x\,,y\,,z)\,,$

   where ${}\left\langle -,-\right\rangle$ denotes the formal evaluation in the sense of the standard inner product.

6. We have

   ${}\left\langle x,x\times y\right\rangle =0=\left\langle y,x\times y\right\rangle \,,$

   where ${}\left\langle -,-\right\rangle$ denotes the formal evaluation in the sense of the standard inner product.

Proof, Write another proof