Dot and Cross Product Division: Is it possible?
2-24-10 to 4-10-10
I have been reviewing statics for a correspondence course on machine design I am taking. The first chapters are basically linear algebra and vectors. While you learn that there are many ways to add subtract and multiply vectors, there are infinitely many results when division is tried. The first thing that went through my mind is "why not?"
You see, back was I was in the third grade the teacher said that it was impossible to divide a larger number by a smaller number. I raised my hand to ask her why and she said "You just can't." I never test the theory to see if you could. I just took her word for it. That was until 5th and 6 grade when we learned fractions. But it took until 10th grade when I learned in class that fractions were just another way of division. But enough of how the public school system quality is itself expressed using very small fractions. Only kidding. My high school math and science teachers were excellent.
OK, so my theory of finding Prime numbers on a parabola or logarithmic spiral then finding when the focus equals a Prime number to find a pattern was off base. Then finding an angle with certain specifications knowing only 2 sides also created a solution which defies logic. And now testing a way to divide vectors is also off base. Why attempt them? The answer is simple: they are interesting. If you do not test the unknown, you only add more information to what is unknown. Testing the unknown is how you learn.
Now the theory.
With using the LSine and LSsine (which are in work previously on www.constructorscorner.net), there may just be a way to determine a unique vector B while dividing the Dot Product (vector C), by a vector A. That is knowing the angle between A and C, the projection of A compared to B might be unique.
This projection is the Ssine (also previously described on www.constructorscorner.net). As C divided by A yields a scalar value which would be the length of B graphically. But we might just find the angle A is from B before it was projected using the cosine.
The LSsine and LScosine may also be useful with the Cross Product and other vector manipulations. The simple part is done: the theory. I challenge the reader to disprove or prove this. The LSine and LScosine also need verified or disprove and useful applications for them are needed.
I will update the site when I have readable math work of this theory of vector division.