The cross product satisfies
[tex]\mathbf i\times\mathbf i=\mathbf j\times\mathbf j=\mathbf k\times\mathbf k=\mathbf 0[/tex]
[tex]\mathbf i\times\mathbf j=\mathbf k[/tex]
[tex]\mathbf j\times\mathbf k=\mathbf i[/tex]
[tex]\mathbf k\times\mathbf i=\mathbf j[/tex]
The cross product is anti-commutative, so that for any two vectors [tex]\mathbf a[/tex] and [tex]\mathbf b[/tex], [tex]\mathbf a\times\mathbf b=-(\mathbf b\times\mathbf a)[/tex], and
[tex]\mathbf j\times\mathbf i=-\mathbf k[/tex]
[tex]\mathbf k\times\mathbf j=-\mathbf i[/tex]
[tex]\mathbf i\times\mathbf k=-\mathbf j[/tex]
It's also distributive, which means we have
[tex](-9\,\mathbf i+2\,\mathbf j-3\,\mathbf k)\times(8\,\mathbf i-3\,\mathbf j+8\,\mathbf k)[/tex]
[tex]=-72(\mathbf i\times\mathbf i)+16(\mathbf j\times\mathbf i)-24(\mathbf k\times\mathbf i)[/tex]
[tex]\,+27(\mathbf i\times\mathbf j)-6(\mathbf j\times\mathbf j)+9(\mathbf k\times\mathbf j)[/tex]
[tex]\,-72(\mathbf i\times\mathbf k)+16(\mathbf j\times\mathbf k)-24(\mathbf k\times\mathbf k)[/tex]
[tex]=(16-9)(\mathbf j\times\mathbf k)+(72-24)(\mathbf k\times\mathbf i)+(27-16)(\mathbf i\times\mathbf j)[/tex]
[tex]=7\,\mathbf i+48\,\mathbf j+11\,\mathbf k[/tex]
