Vector addition is based around the trigonometric attributes around the triangle, thus the gnomon in the unit circle is very instructive.
The gnomon in the unit circle ties together the three sides of a triangle and the associated arcs. In vector terms any two vectors when added rotate and extend or diminish each other . Thus addition is effectively a rotation transformation.
The summation of vectors is defined geometrically around a form and it is an action. The action is to get from A to B. Why aggregate around a form? The behaviour of space and the motion in space respects form or deforms form. Respecting form means we have to aggregate distances and going around a perimeter does that. Deforming space means the lengths or edges have to be elastic and defining a unit vector with a scalar represents that. So noting the form produced y ana ction records all these conditions. So why highlight one dimension as the resultant when the geometric form contains the information?
When i perform the action of measuring or traveling my interest is the direct measuremnt, but i do not want to lose the other information. so the resultant vector is not always the goal.The systematic application of operations produces a record, which can reveal combinatoric information and derive formulae. Vectors begin to highlight effective organisation of the permutation and combination of actions in space.
Suppose we concentrate on the resultant, then addition traces an extension and then a combined rotation and extension. This describes well the effect of non parallel forces acting on the same object. Thus the form we trace out by drawing motion holds true for the motion of larger objects. The trace records motion under the same type of force distribution scaled down. Thus the vector trace also records model forces, and suggests applicability. The geometry in this case is secondary to the force model used to trace the geometrical figure.
Using this analysis we can define a rotation vector that represents a drawing force that rotates a centre. Again the motion can be characterised as an extension followed by a extension around the circumference of a circle. In this translation there is no further radial extension. Not shown usually in drawing the circle is the vector triangle that is in a pair of compasses, or the tension vector in a taught string anchored at the origin. or the action and rection vectors as one draws around a circular form, Every one of these vectors needs to be recorded to describe the actual situation. When that is done then the proper vector description can be noted and the proper geometrical form results which gives the desired results. We can also se how a rotational unit vector is interrelated to other vectors in the situation.
In one case vector addition is the same as just an extension, when the vectors are parallel or contiguous, but as i have said before position of a vector has also to be recorded otherwise the missing positional information can lead o a distorted result.
When i look at the analogical frame of references that encapsulate the analogies vectors are linked to i see that vectos as extensions of forms are used to model motion But we have no direct way of using motion as an analogical basis in our current systems. The tautology always starts with extensions of space or form, and we represent motion by a trace which is an extension of form called a trace line or a locus.
The only pure way to capture motion is in sequents and these seqoents have to be ordere and displayed exactly as in a film sequence or a series of frames. Motion sequents are the only analogical base that uses motion not extension, abd i will continue to explore its impact on describing reality.
The research question is using the definition of a motion sequent as an image frame in a fim reel how do i devise a notation system to describe what is happening in sequence of frames. How do i generalise this to 3d, and how do i directly relate this to electrnic storage of film sequences? Do the different forms of recording alter the nascent algebra ? Does the media editing industry already have a model i can use and generalise? What is the significance of the computing paradigm and the influence of computer science? Has computer science and programming languages already developed an algebra in flow charting? Does iteration and convolution have a role in such an algebra? Does fractal geometry< and the work of Benoit Mandelbrot?
The possible answer is out there. I can feel it in my flowing water!