Back to the Rotation Matrices

Here I am interested in playing with the idea of two vectors, \(\vec{u}\) and \(\vec{v}\), which define a three dimensional space, enumerated by reals \(\mathbb{R}^{3}\). I am also interested in playing with their rotational transformations and further progress the discussion in my previous post.

To this end, I will look at how to define general rotation matrices for a third unit vector, \(\hat{B}_{\parallel}\), that bisects these two original vectors. I will also look at how a three dimensional coordinate system may be constructed by defining two additional unit vectors, \(\hat{B}_{\perp}\) and \(\hat{B}_{\hat{n}}\), that result from operating on the bisecting vector with the generally defined rotation matrices.

I will start by making the following assumptions:

1. The space defined by the two vectors, \(\vec{u}\) and \(\vec{v}\), is locally euclidean in the three dimensional space they define.
2. The unit vector \(\hat{B}_{\parallel}\) can be expressed by the column matrix \(\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}\).

The second point implies the existence of an additional two or more linearly independent vectors which can be described by similar column vectors. 

The reason I am restricting the discussion to three dimensions is due to the aparent observation (which may be true or not) that we live in a space which is three dimensional and, as a result, I am interested in working in three dimensions. For now, I will leave the discussion of dimensional spaces greater than three to the following statement: One can use similar ideas as the ones presented herein to construct higher dimensional spaces. 

In the future, and for more playful reasons, I will be interested in the spaces between seven and eight dimensional space. This is due to the fact that, for a locally euclidean n-dimensional space, the volume of a sphere (n-sphere) in that space is maximized somewhere between seven and eight dimensional space:

$$V_n \left(R \right)=\frac{\pi^{ \frac{n}{2}}}{\Gamma \left(\frac{n}{2}+1 \right)}R^{n}$$

Above, \(\Gamma \left(\frac{n}{2}+1 \right)\) is the gamma function.

Ok, now to get started!

Say that there are two intersecting line segments, of length \(\left\|u\right\|\) and \(\left\|v\right\|\), somewhere in n-dimensional space at an angle \(\theta\) from one another.

Bisecting the angle \(\theta\) and defining a new line segment along the angle of bisection \(\alpha\) (I call these derived virtual line segments when working on physics problems), we can consider that the original two line segments \(\left\|u\right\|\) and \(\left\|v\right\|\) have some sort of direction relative to  each other at the point of intersection.

... more to come later ...