It is symmetric in x and y: x ⋅ y = y ⋅ x. It is linear in its first argument: (ax1 + bx2) ⋅ y = ax1 ⋅ y + bx2 ⋅ y for any scalars a, b, and vectors x1, x2, and y. It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0.