The application of "vectors" is not as straight forward as it is made out to be.
The Strecken arose in Grassmann's mind through considering Dynamic Mechanical principles. and their geometrical representation. A long mechanical history of applying geometrical diagrams to solve and represent dynamic situations underpins the notions used to denote force and velocity and momentum, but it took Newton and Moebius and a few others to tease out an algebra in these diagrams. Hamilton and Grassmann both contributed to the notation, but Grassmann expounded it the deepest.
However, Gibbs hijacked Grassmann's Strecken and misapplied it, in a battle to the death with Hamilton's Quaternions. In the aftermath many have been confused about the application of vectors/strecken.
The mechanical application is not the same as the mathematical. The analysis and the notation has to be accurate if the method is to be applied accurately. On top of that, the method has to be fudged creatievly to give the "right" answers .
This fudging is very revealing, even instructive; it provides additional insights into the working of "reality" and allows modification of the method to accomodate. This is where Gibbs went wrong: Grassmann viewed his strecken as a tool for analysis , not a description of reality, not a secret sinuous skeleton of reality.
Grassmann constantly modified his tools, sharpened them, honed them for better use.
Inertial properties of space naturally evoke a representation by a line symbol. Inertial momentum again is well symbolised by a line. Billiard balls in their interactions, traced out vector paths and geometrical lines, diagonals of parallelograms. Spinning tops demonstrated angular momentum and the combination of angular momentum and linear momentum provided Newton with intriguing insight into the motion of the planets in "heaven".
In a fmous note on the motion of bodies Newton explores the parallelogram rule for adding "vectors", in terms of compounding tangents, as in the method of compound interest.
Grassmann took a different insight based on 3 points and the law of 2 Strecken. He took a different meaning to the parallelogram, and it is this confusion that Gibbs passed on. Hamilton was not clearer, because his notation was offputting. It took the invention and innovation of others to meld from the 2 the dot product, the cross product, the wedge products,etc., From Grassmann the exterieor and interior products. These all provide a powerful vector notation for describing abstractly dynamic situations.
The notation is used to describe many complex systems of equations describing complex behaviours and multiple simultaneous situations. The vector, quaternion notation has become standard, along with the Bra Ket, and other Quantum Mechanical notation. These different Algebras use many tools to hold the information together, in a manageable way, but they all now rely on vectors and multiplexes of vectors called matrices and Tensors.
These tools are so ubiquitous they are almost the medium itself, and that is why it is important to understand that they are tools of analysis for the subjective experience we call "reality". This reality is based on the inertial properties of space, particularly angular and linear momentum.