3-vectors are triplets of real numbers that transform under rotations like the coordinates 4-vectors are quadruplets of real numbers that transform under Lorentz transformations like the coordinates of
You will remember that the scalar product of two 3-vectors is invariant under rotations of the (spatial) coordinate axes; after all, this is why we call it a scalar. Similarly, the scalar product of two 4-vectors and defined by
is invariant under Lorentz transformations (as well as translations of the coordinate origin and rotations of the spatial axes). To demonstrate this, we consider the sum of two 4-vectors and calculate
The products and are invariant 4-scalars. But if they are invariant under Lorentz transformations, then so is the scalar product
One important 4-vector, apart from is the 4-velocity which is tangent on the worldline is a 4-vector because is one and because is a scalar (to be precise, a 4-scalar).
The norm or "magnitude" of a 4-vector is defined as It is readily shown that the norm of equals (exercise!).
Thus if we use natural units, the 4-velocity is a unit vector.