Kurs:Mathematik für Anwender (Osnabrück 2019-2020)/Teil I/Repetitorium/22/Skalarprodukt/Studentenfrage/Antwort
I'm not sure if I understand your question. The german translation for dot product is Skalarprodukt and it is only briefly mentioned in this lecture and only really introduced in the second semester course. Why the dot product is useful is a question better suited for that course.
It is important not to confuse the terms dot product (or Skalarprodukt) and scalar multiplication (or Skalarmultiplikation). The former is an important operation on two vectors and gives a scalar, while the latter is one of the two fundamental operations on a vector space which multiplies a scalar and a vector to get a multiple of the vector. For us in this semester only the scalar multiplication is important.
The scalar multiplication is really fundamental to vector spaces and without it we wouldn't have any structural relationship between the underlying field and the vector space. In that sense it is more a structural necessity for meaningful linear algebra than a tool for computations.