To the physicist, their notation simplifies and generalizes concepts first set in motion by Newton. In the above it is immaterial whether p and q are vector or scalar variables. O nullity 6. All the above operate upon vectors and yield vectors, generalizing the algebra of scalars.
Vectors can multiply in a number of ways.
The dot product is formed by pairwise multiplying the components of the two vectors and summing. The scalar result may be positive, zero or negative. By this linear algebra is related to trigonometry. Without tools of vector analysis, solution would be cumbersome. Dotting a vector with itself gives that vector's squared length.