Vector product has cross multiplication and point multiplication, so the multiplication symbol of vector product can't be omitted, and so far, I haven't seen the writing of vector a square, either A A, a×a or the square of | A |.

According to the meaning of the question, we can find that the square of a in the question refers to the square of | a |, but unless the meaning of square is agreed in advance, this writing is strongly recommended.

In physics and engineering, geometric vectors are more often called vectors. Many physical quantities are vectors, such as the displacement of an object, the force exerted on it by a ball hitting a wall and so on. On the contrary, it is scalar, that is, a quantity with only size and no direction. Some definitions related to vectors are also closely related to physical concepts. For example, vector potential corresponds to potential energy in physics.

The concept of geometric vector is abstracted from linear algebra to get a more general concept of vector. Here, a vector is defined as an element of a vector space. It should be noted that these abstract vectors are not necessarily represented by number pairs, and the concepts of size and direction are not necessarily applicable. Therefore, it is necessary to distinguish the concept of "vector" in the text according to the context when reading on weekdays.

However, we can still find the basis of a vector space to set the coordinate system, and we can also define the norm and inner product on the vector space by choosing a suitable definition, which enables us to compare abstract vectors with specific geometric vectors.