orthogonality

In a two-dimensional context, if the vector (x₁,x₂) is orthogonal to (x₁,x₂), we would have x₁y₁+x₂y₂=0, or x₂/x₁ = - y₁/y₂ and the vectors would have negative reciprocal slopes. The minus sign in the Minkowski version gives them reciprocal slopes, making them mirror images in the diagonal. On the diagonal itself, the light cone, vectors are self-orthogonal, the same as being null. In three dimensions the diagonal is actually a cone, to which the orthogonal plane is tangent. Planes Minkowski-orthogonal to axes of other inclinations are filled with vectors of the opposite timeliness or spaceliness from the axis itself. That would seem to imply that the vector product of vectors of opposite timeliness would be null, but things are a little more complicated than that.

The triple vector product is not associative, but is multilinear in each of its factors. Again in analogy to the Euclidean formula,

$$({\bf u}\times {\bf v}) \times {\bf w} {\bf u}({\bf v},{\bf w}) - {\bf v}({\bf u},{\bf w})$$ = (58)