The principle of superposition of magnetic fields, as well as any other principle of superposition, is based on vector essence of the field of magnetic induction. It allows to simplify finding of value of magnetic field in any point.
Vector magnetic field
So, magnetic field – this vector field. It means that in each point of space this field forms a vector, and not just some scalar value. That is magnetic field in any point of space works in a certain direction. Thus, it is possible to set a set of the directed pieces forming the field. If to represent graphically such field, then it will represent big (or even infinite) quantity of the vectors forming the uniform vector field.
Peak-a-boo property of vectors of magnetic field
If magnetic field is a vector, then all properties of vectors have to be applicable to it. One of the most important properties of vectors which even defines the concept of the directed piece is the possibility of summation of vectors. That is, if there are, say, two vectors, then there is always the third, being the sum of the first two vectors.
In this case it is about vectors of magnetic field. Therefore it is supposed to summarize vectors of magnetic induction, and the sum is understood as the full or peak-a-boo field with which it is possible to replace a set of fields of its components. Thus, the principle of superposition says that induction of the magnetic field created by several sources is equal in this point of space to the sum of the magnetic fields created by each of sources separately. Now it becomes clear that the vector sum of fields is supposed. It is important to notice that mean not the sum of vectors of this vector field, but the sum of vectors of various vector fields created by various sources, but in one point. This principle gives the chance to count magnetic fields in difficult situations incredibly simply. Knowing what distribution of magnetic field of any elementary sources (the conductor with current, the solenoid, etc.), it is possible to design any necessary magnetic field which field can be calculated from such simple elements, using the principle of superposition of magnetic fields. Biot-Savart-Laplace's law is the most important consequence of the principle of superposition of magnetic fields. This law generalizes the principle of superposition in case of the infinitesimal, making the full field vectors. Summation in this case is replaced with integration on all infinitesimal vectors of magnetic induction. Such elementary vectors of induction usually are currents of conductors. Thus, integration (summation) is conducted on all length of the conductor on which current flows.