Learn something new every day. The dot product is a tensor, it maps product of two vectors. I always thought of it simplistically as multiple dimensions. A generalization where 2d and 3d vectors are a simplified cases.
https://en.wikipedia.org/wiki/Tensor
In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can take several different forms, for example, a scalar, a tangent vector at a point, a cotangent vector (dual vector) at a point, or a multi-linear map between vector spaces. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors.[1] While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system.
An elementary example of mapping, describable as a tensor, is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor, because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol ε i j k {\displaystyle \varepsilon _{ijk}} {\displaystyle \varepsilon _{ijk}} nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.
Definition[edit]
Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.