Draft:Linear algebra/Rotation of axes

- Notice: Tensors/Transformation rule under a change of basis gets about 200 pageviews per month. This draft might need to be moved, or at least linked. - Linear algebra/Rotation of axes, Rotation matrix, Rotation of axes

These days most introductory physics textbooks display the position (displacement) vector as:

"$\vec r = x\hat i + y\hat j + z\hat k$"

This use of seven letters is peculiar in a field where the combined Latin, Greek, and Cyrillic alphabets sometimes seems insufficient. Here, we shall replace the unit vectors $$\{\hat i, \hat j, ...\}$$ by $$\{\hat x, \hat y \ldots \}$$, $$\{\hat e_1, \hat e_2, \ldots \}$$, or even simply $$\hat e_j$$, where $$j=\{1,2,3\}$$. Even though matter exists in three spatial dimensions, one dimension can often be "ignored". The two-dimensional figure shown can also model rotations rotations of a cube or long wire of uniform cross sectino. Two different coordinate systems can be used to represent the point $$P$$:"$\underline x = \sum_j x_j\hat e_j=\sum_j x^\prime_j\hat e_j^\prime$"We shall see that underline convention for vectors is also useful for tensors. The unit vectors reminds readers that the coordinate system has been rotated.

- Quizbank ideas: ''Does the value of $$\underline x$$ change if the index $j$ is changed? Can the figure be applied to a three dimensional object? Why is $$\underline x^\prime$$ absent from this discussion? Is $$\hat x\cdot\hat y^\prime$$ positive, negative, or zero? Though it might seem trivial, the following question serves a purpose:'' It is the author's opinion that three dimensional objects should generally be drawn in perspective by displaying the x,y, and z axes (True or False)? -

Deriving the rotation tensor
The dot product of any vector with a unit vector yields the vector's component in that direction. Pick any value for $$j$$ and consider the dot product with and vector consider for example the dot product of $$\underline x$$ with the unit vector $$\hat e_k$$:

$$\hat e_k\cdot \underline x= \hat e_k\cdot\sum_j x_j\hat e_j = \sum_j \underbrace{(\hat e_k\cdot\hat e_j)}_{\delta_{kj}=\text{0 or 1}}\; x_j= \sum_j x_j\delta_{jk}=x_k,$$

where $$\delta_{jk}$$ is the Kronecker delta function. Here we took the dot product with the unprimed unit vector $$\hat e_k$$. Something entirely different happens if we instead multiply by the primed unit vector $$\hat e_k^\prime$$:

$$\hat e_k^\prime\cdot \underline x= \hat e_k^\prime\cdot\sum_j x_j\hat e_j = \sum_j (\hat e_k^\prime\cdot\hat e_j)x_j \;\equiv\; \underline\underline R \cdot \underline x,$$ where, $$R_{jk}=\hat e^\prime_k\cdot\hat e_j$$, $$\,\left(\text{in dyad notation, } \underline\underline R= \hat e^\prime_k \hat e_j\right)$$

is the rotation tensor. To understand why, note that since $$\hat e_k^\prime\cdot \underline x=x_k^\prime,$$ we have an expression for the k-th component of $$\underline x$$ in the primed $$(x_1^\prime,x_2^\prime\ldots)$$ coordinate system.

$$ \hat e_k^\prime\cdot \underline x=x_k^\prime \; \Rarr \; \sum_j\hat e_jx^\prime_j = \sum_{j,k}\hat e_jR_{jk}x_k \; \Rarr \; x^\prime_j = \sum_kR_{jk}x_k,$$

The latter form can be written as $$x^\prime_j = R_{jk}x_k$$ if it is understood that repeated and adjacent subscripts are always summed.

Scratchwork
Our first tensor will of course be the rotation tensor: $$x^\prime_j = \sum_k R_{jk}x_k.$$

To appreciate the value of unit vectors, we calculate the rotation tensor for the two dimensional rotation shown above. Let $$\underline x$$ be the displacement vector from the origin to point $$P.$$

$$\underline x = \sum_j x_j\hat e_j = \sum_j x_j^\prime\hat e_j^\prime,$$

where, for example, $$x_2^\prime = y^\prime,$$ in the figure above. Interesting results emerge when we take the dot product of this with either $$\hat e_k$$ or $$\hat e^\prime_k,$$ because for unit vectors, <math since="" we="" are="" dealing="" with="" unit="" vectors,="" the="" identity,="" $$\underline A\cdot \underline B = AB\cos\theta,$$ leads to:

$$\begin{array}{c|c} \hat e_1\cdot\hat e_1^\prime = \cos\theta & \hat e_1\cdot\hat e_2^\prime=-\sin\theta \\ \hline \hat e_2\cdot\hat e_1^\prime= \sin\theta & \hat e_2\cdot\hat e_2^\prime =\cos\theta \end{array}$$

$$\begin{array}{l|l} \hat e_1\cdot\hat e_1^\prime = \cos\theta \quad & \quad \hat e_1\cdot\hat e_2^\prime=-\sin\theta \\ \hline \hat e_2\cdot\hat e_1^\prime= \sin\theta \quad &\quad \hat e_2\cdot\hat e_2^\prime =\cos\theta \end{array}$$

If find writing this as, $$\underline x^\prime = \underline\underline R \cdot \underline x,$$ to be simultaneously vague and useful. To understand why, we "derive" the rotation tensor $$\underline\underline R$$ for the two dimensional graph shown above. Let

The great virtue of the unit vector is that it's dot product of any vector with a unit vector reveals a component of that vector. The tensor subscript notation permits us to "vaguely" take the dot product with each unit vector in the rotated reference frame, i.e., with $$\hat e^\prime_k$$:

(depending on whether $$j={1,2}$$ or $$j={1,2,3}$$.) The underline convention will facilitate the introduction of tensors:

start
$$\begin{array}{rcl} \underline a\cdot\hat e^\prime_k & = & \sum_j a_j\hat e_j\cdot\hat e^\prime_k \\ a^\prime_k & =&  \sum_j a_j\hat e_j\cdot\hat e^\prime_k \\ & =& \sum_j (\hat e^\prime_k\cdot\hat e_j) a_j\\ \end{array}$$

$$\begin{array}{rcl} \underline a\cdot\hat e^\prime_k & = & \sum_j a_j\hat e_j\cdot\hat e^\prime_k \\ a^\prime_k \quad & =& \sum_j \;a_j\hat e_j\cdot\hat e^\prime_k \\ a^\prime_{\begin{matrix} k &\\ \downarrow &\\ j \end{matrix}}  & =& \sum_{\begin{matrix} j &\\ \downarrow &\\ k \end{matrix}} (\quad\hat e^\prime_{\begin{matrix} k &\\ \downarrow &\\ j \end{matrix}}                         \cdot\quad\hat e_{\begin{matrix} j &\\ \downarrow &\\ k \end{matrix}} ) \quad a_{\begin{matrix} j &\\ \downarrow &\\ k \end{matrix}} \\ \end{array}$$