What is inner product and dot product?

Question

Linear algebra

I take courses in higher mathematics in English, the terminology is a little confusing.
As I understand it, there is a dot and cross product of vectors, depending on the result that we want to get. There is also matrix multiplication.

In English, I heard two terms dot product and inner product. The first is a special case of the second and is described by the formula:
<x,y> = x.T @ I @ y , где I - единичная матрица

But what is it called in Russian? And what is the geometric meaning of these operations?

Answer 1

Dot product is a special case of inner product, for example in Euclidean vector space. We can say that the dot product is a special inner product defined in the R^n space.
The geometric meaning of the dot product is related to the projection of vectors onto each other. That is why for orthogonal vectors a and b, (a, b) = 0.
I don’t know why you mentioned the cross product here, this is generally from a different opera and does not apply to the title of the question. It is defined only for three-dimensional Euclidean space (although there is also a pseudo vector product for two-dimensional space). The geometric meaning is such that the result of the cross product of vectors a and b is vector c, perpendicular to both vector a and vector b at the same time.

Answer 2

As far as I know, both concepts are a dot product. Multiplication by the identity matrix has no practical meaning.