In case you’re not familiar with it, the cross product is very handy in 3d geometry:

aXbis a vector perpendicular toaandbin direction given by right hand rule (rotate (screw)aintobthru the acute angle θ between them), and magnitude absinθ.

aXb= “det” |i,j,k; a1, a2, a3 ; b1, b2, b3| , = |row; row; row|, (not obvious, proof in textbooks)

Ex (0,0,1) X (a1,a2,a3) = |i,j,k; 0,0,1 ; a1,a2,a3 | = -a2i+ a1j= (-a2,a1,0)

It follows thata.bXc= det |a;b;c|

i,j,kis orthonormal basis (also callede1,e2,e3). "det" not really determinant- write out as determinant and evaluate as determinant.

It would be visually clearer if I could print determinants, sorry.