Vectors and the Laws of Physics — Question 6

Question 6

An isotropic scaling law has the same form in every orientation

A device obeys $\vec B=\lambda\vec A$ with dimensionless $\lambda=-2$. For one input, $\vec A=(3,-1)$ m. Assume the relation applies to all inputs and the same physical vectors are redescribed in axes turned $90^\circ$ counterclockwise.

1. Find B in the old frame and both vectors in the new frame.

2. Check whether the same coefficient relates both new components.

3. Explain what would follow for any rotation angle, and why this coordinate check alone cannot prove the device law experimentally.

Question 6 - Solution

An isotropic scaling law has the same form in every orientation

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) Evaluate and transform.

$$\boxed{\vec B=(-6,2),\quad\vec A'=(-1,-3),\quad\vec B'=(2,6)}\quad\mathrm m$$ $\vec B=(-6,2)$ m. The passive quarter-turn mapping $(x,y)\mapsto(y,-x)$ gives $\vec A\,{}^{\prime}=(-1,-3)$ m and $\vec B\,{}^{\prime}=(2,6)$ m.

(b) Apply the scalar equally in every direction.

$$(-2)\vec A'=(-2)(-1,-3)=(2,6)\,\mathrm m=\vec B'$$ $B_{x\prime}=(-2)(-1)=2$ m and $B_{y\prime}=(-2)(-3)=6$ m. The coefficient remains $-2$; the new components again describe a vector twice as long and opposite in direction to A.

(c) Separate mathematical consistency from empirical truth.

$$Q_\phi(\lambda\vec A)=\lambda Q_\phi\vec A\Rightarrow\vec B'=\lambda\vec A'\quad\text{for every rotation }Q$$ Every rotated component is a linear combination of old components, so multiplying all of them by the same scalar commutes with the coordinate transformation. The law retains its form for any rotation. That proves consistency under redescribing the assumed relation; measurements are still needed to establish whether the device actually obeys it for other inputs.