K^3/Cross product/Properties/Fact

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

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

and

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

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

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

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

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

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

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

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