In this paper we firstly use a new method--the invariance of space-time interval and some simple linear algebra knowledge to derive Lorentz transformations and four-dimensional vectors. Finally we discuss and prove how to define the force and the momentum in relativity which has not been discussed and proved in textbooks and scientific literature. The first three dimensions of a four-dimensional momentum are defined as momentum and the derivative of momentum with respect to time is defined as force. But there is a problem that the rationality of the definition of momentum is not discussed and proved. Force and momentum cannot be arbitrarily defined. Because if our senses are sensitive and sophisticated enough, only a correct definition can guarantee that when we accelerate an object with a constant force, the momentum will increase at a constant rate. It is not necessary to be discussed in classical mechanics, because in classical mechanics the force is proportional to the acceleration and the force comes before the momentum. But it is just the opposite that the momentum comes before the force in relativistic mechanics, so it's important to discuss and prove how to define the force and the momentum in relativity. In addition the fact that the same physical process does not depend on the space-time point means that the Lorentz transformations must be linear transformations, so we can derive Lorentz transformations and four-dimensional vectors by using the invariance of space-time interval and some simple linear algebra knowledge.