Prove that `|a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot`

Explanation

Let
` Delta=|{:(ax-by-cz,,ay+bx,,cx+az),(ay+bx,,by+cz-ax,,bz+cy),(cx+az,,bz+cy,,cz-ax -by):}|`
Applying `C_(1) to xC_(1) +yC_(2)+zC_(3),` we get
` Delta =(1)/(x) |{:(a(x^(2)+y^(2)+z^(2)),,ay+bx,,cx+az),( b(x^(2)+y^(2)+z^(2)),,by+cz-ax,,bz+cy),(c(x^(2)+y^(2)+z^(2)),,bz+cy,,cz-ax -by):}|`
Taking `(x^(2)+y^(2)+z^(2))` common from `C_(1)` and then applying `R_(1) to aR_(1) +bR_(2)+cR_(2)`
` Delta =((x^(2)+y^(2)+z^(2))/(ax))xx`
`|{:((a^(2)+b^(2)+c^(2)),,y(a^(2)+b^(2)+c^(2)),,z(a^(2)+b^(2)+c^(2))),( b,,by+cz-ax,,bz+cy),(c,,bz+cy,,cz-ax -by):}|`
Taking `(a^(2) +b^(2)+c^(2))` common from `R_(1) ` and then applying `C_(2) to C_(2) -yC_(1)" and " C_(3) to C_(3)-zC_(1)` we get
`Delta =((x^(2) +y^(2) +z^(2))(a^(2)+b^(2)+c^(2)))/(ax)xx |{:(0,,0,,0),( b,,-cz-ax,,cy),(c,,bz,,-ax-by):}|`
Expading along `R_(1)` we get
`Delta =(1)/(ax)(x^(2) +y^(2) +z^(2)) (a^(2) +b^(2)+c^(2))xx`
`[aczx +bczy+ a^(2)x^(2)+abxy -bcyz]`
`=(x^(2)+y^(2)+z^(2))(a^(2)+b^(2)+c^(2))(ax+by+cz)`