discussion of spacetime. Definition of 4-vectors A 4-tuple, is a "4-vector" if its component Ai transform between frames according to the Lorentz transformation. If using coordinates, A is a if it transforms (in the ) according to which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation. As usual, when we write x, t, etc. we generally mean Δx, Δt etc. The last three components of a must be a standard vector in three-dimensional space. Therefore, a must transform like under Lorentz transformations as well as rotations. Properties of 4-vectors Closure under linear combination: If A and B are , then is also a . Inner-product invariance: If A and B are , then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a . In the following, and are : In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in . Two vectors are said to be orthogonal if Unlike the case with orthogonal are not necessarily at right angles with each other. The rule is that two are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike is orthogonal with itself. Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval and the invariant length of the relativistic momentum vector Examples of 4-vectors Displacement 4-vector: Otherwise known as the spacetime separation, this is or for infinitesimal separations, . Velocity 4-vector: This results when the displacement is divided by , where is the proper time between the two events that yield dt, dx, dy, and dz. The is tangent to the world