A vector has only two concepts: length (size) and direction. When we need to calculate the area, we need two vectors, in other words, two vectors that are not parallel produce the length, width and area (and of course the direction), but when we need to study the three-dimensional problem, we design three directions, and we need one vector, which constitutes most of the solids we see.

The generation of vectors is introduced in the process of studying problems. We know that for two nonparallel vectors, they are irrelevant and unrepresentable, but through operation, they can represent any vector or even area on the plane. In practice, they can represent the result of the interaction between one vector and another, such as work and power, which is the dot product. In the three-dimensional case, these two vectors have nothing to do with the vectors on the other plane, but in practical research problems, there are many situations that need to represent the vectors on the other plane, such as the direction of force, linear velocity and angular velocity. Often this vector is perpendicular to the two vectors on the plane (just what we have learned at present), so in order to make the two vectors on the plane represent the other vector conveniently, we introduce the cross product, that is, cross product, where the direction perpendicular to the two vectors represents the direction of the other vector, and the size is determined by the size and included angle of the two vectors. So the mixed product (dot product and vector product) is used to represent the volume.