In vector algebra there is a tendency to confuse the dot product with the magnitude of a vector. It is the square of the magnitude of a vector, or the magnitude of the projection onto a unit vector, or the scalar of a vector in any direction different to a referent vector. Thus the dot product stretches or diminishes the magnitude of any vector in a relative orientation to a referent vector. The dot product acts on the magnitude of a vector as a scalar,it is not a vector .
The product rules for "multiplying" combinations are in fact combination tables, not multiplication tables. They show the combinatoric bonds in tabular form. When applied to number bonds, only one form of the tabular presentation may be called multiplication , The table may equally well represent bonds for Addition ,division and subtration, as well as translation rotation reflection and magnification etc. They are combinatoric devices sometimes presented in "tree" form.
Thus before any meaning can be attached to bonds, the possible bonds are laid out in this way, and then the author applies definitions to their connection. These definitions are subjective, and represent the labour of much thought often, to obtain consistency. However, sometimes they are ill conceived and in need of correction. However in many cases a small change suffices to reveal a world of applicability.
These combinatoric tools a re part of the philosophers set of tools by which to reconstruct a form after it has been analysed into monads of appropriate magnitude and form. Our subjective tools should not be confused with objective reality, but should assist us to interact with reality in a cybernetic way.
I still have to be careful, because + – * / sign and i, have become firmly ensconced as part of the folklore of mathematics. The combinatorial processes they refer to are subliminal to most mathematicians i for example refers to the process of finding the geometric mean in a unit circle; few if any ever give that any consideration.
i has had a confusing history since Euler denoted it. Bombelli actually identified i as an adjugate magnitude,but called it pui di meno, Descartes denoted them as imaginary in a literal sense, and in context of polynomial theory important, but ridiculous in the context of heing of any use. Euler when considering the measurement of arcs , particularly in spherical geomtry used i to stand for infinity. This was to stand for the ray used to calculate the sine. He recognised it as a magnitude but imaginary , that is not commensurable, but nevertheless obeying the rules i = -1/i. It was Gauss that combined the different magnitudes in a linear combination a + ib, and in this way vindicated Bombellis insight, that i was some kind of magnitude. Thus Gauss Hamilton and Grassmann developed an algebra od linearly combined magnitudes.
However Hamilton developed the quaternions, and introduced the name vector for a part of a quarenion. Gibbs wanted to use the name vector for an oriented quantity, and using the combinatorial tables developed by Grassmann he found that the same results could be obtained regardless of i, simply by the rules of combination. I had no role as a imaginary magnitude in his conception. it was merely a vector orientation. However he could not get his poin across becaus Hamilton and the Quaternion sect insisted on i being a magnitude with imaginary powers!. It is no wonder that Gibbs declared that Quaternions should be "murdered". The confusion was now complete i was a number, a magnitude a vector, and some kind of rotation.
For me i is a vector direction, a special one as is j and k. amd the vector algebra enshrines how vectors are combined. In addition we need to remove the notion that sign is anything more than a special π rotation of a vector, and replace sign by a vector of orientation.
We can see that the processes we think of a basic operators are then spatial transformations Translation and rotation of object. We can extend that tho deformation and magnification. Reflection in a centre of rotation is the only tool that measures abd deals with subjective processes in the main.