In case you’re not familiar with it, the cross product is very handy in 3d geometry:
aXb is a vector perpendicular to a and b in direction given by right hand rule (rotate (screw) a into b thru 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 that a.bXc = det |a;b;c|
i,j,k is orthonormal basis (also called e1,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.