<P> Because of this, two vectors satisfying ⟨ x, y ⟩ = 0 (\ displaystyle \ langle \ mathbf (x), \ mathbf (y) \ rangle = 0) are called orthogonal . An important variant of the standard dot product is used in Minkowski space: R endowed with the Lorentz product </P> <Dl> <Dd> ⟨ x y ⟩ = x 1 y 1 + x 2 y 2 + x 3 y 3 − x 4 y 4 . (\ displaystyle \ langle \ mathbf (x) \ mathbf (y) \ rangle = x_ (1) y_ (1) + x_ (2) y_ (2) + x_ (3) y_ (3) - x_ (4) y_ (4).) </Dd> </Dl> <Dd> ⟨ x y ⟩ = x 1 y 1 + x 2 y 2 + x 3 y 3 − x 4 y 4 . (\ displaystyle \ langle \ mathbf (x) \ mathbf (y) \ rangle = x_ (1) y_ (1) + x_ (2) y_ (2) + x_ (3) y_ (3) - x_ (4) y_ (4).) </Dd> <P> In contrast to the standard dot product, it is not positive definite: ⟨ x x ⟩ (\ displaystyle \ langle \ mathbf (x) \ mathbf (x) \ rangle) also takes negative values, for example for x = (0, 0, 0, 1) (\ displaystyle \ mathbf (x) = (0, 0, 0, 1)). Singling out the fourth coordinate--corresponding to time, as opposed to three space - dimensions--makes it useful for the mathematical treatment of special relativity . </P>

Q is not a vector space over r