It's actually vx^2 + vy^2 + vz^2 - vt^2 the minus sign in front of the vt is very important and gives the SR (hyperbolic) structure of flat spacetime. The r

It's actually vx^2 + vy^2 + vz^2 - vt^2 the minus sign in front of the vt is very important and gives the SR (hyperbolic) structure of flat spacetime. The rotating a vector thing is just an analogy to euclidean space rotations.