Vectors and the Laws of Physics — Question 7

Question 7

A directional response needs its material axes

In axes fixed to a sheet, a device maps a displacement-like input to an output by $B_x=2A_x$ and $B_y=3A_y$, both in metres. Take $\vec A=(1,1)$ m. An observer rotates the coordinate axes $90^\circ$ counterclockwise without turning the sheet.

1. Transform the actual input and output to the new frame.

2. Test the claim that the unchanged component formulas $B_{x\prime}=2A_{x\prime}$ and $B_{y\prime}=3A_{y\prime}$ still apply.

3. Write the correct formulas in the new frame and explain whether the failure of the unchanged formulas makes the physical response invalid.

Question 7 - Solution

A directional response needs its material axes

Here $Q_\phi$ transforms components when axes turn counterclockwise by $\phi$; $R_\phi=Q_{-\phi}$ rotates the arrow in fixed axes. An unsubscripted $Q$ denotes the same component transformation, and $I$ is the identity.

(a) Transform the physical output, not just the input.

$$\boxed{\vec A'=(1,-1),\quad\vec B'=(3,-2)}\quad\mathrm m$$ The sheet produces $\vec B=(2,3)$ m. The new pairs are $\vec A\,{}^{\prime}=(1,-1)$ m and $\vec B\,{}^{\prime}=(3,-2)$ m.

(b) Detect the misplaced directional coefficients.

$$\vec B'_{\mathrm{wrong}}=(2A_x',3A_y')=(2,-3)\,\mathrm m\ne(3,-2)\,\mathrm m$$ The unchanged formulas predict $(2,-3)$ m, different from the actual $(3,-2)$ m. The factor 2 belongs to the original sheet x direction, not to whichever direction an observer happens to call x.

(c) Carry the material orientation into the description.

$$M'=Q\begin{pmatrix}2&0\\0&3\end{pmatrix}Q^{-1}=\begin{pmatrix}3&0\\0&2\end{pmatrix},\quad B_x'=3A_x',\quad B_y'=2A_y'$$ Here $Q=Q_{90^\circ}$ and $M$ denotes the response matrix. The correct new formulas are $B_{x\prime}=3A_{x\prime}$ and $B_{y\prime}=2A_{y\prime}$. The response is anisotropic: it distinguishes two physical sheet directions. This is a valid coordinate-independent physical model when its material axes are included. It is the naive claim of identical diagonal coefficients in every orientation that fails.