[QUE/ME-02001] ME-PROBLEM

$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
Show under an infinitesimal rotation of a vector \(\vec{A}\) about an axis \(\hat{n}\) has the form \(\Delta \vec{A}= c \hat{n}\times{\vec{A}}\), where \(c\) is an infinitesimal constant. }
\Hint{Use the fact that lengths and angles do not change under rotations} \QField{Comment}{ When ever you see an equation of the form \(\Delta \vec{A}= c \hat{n}\times{\vec{A}}\), you should realize that some rotation is taking place. Several examples can be given

1. A electric dipole placed in an electric field will experience a torque and its equation of motion can be written down as\[ \dd[\vec{p}]{t}\propto \vec{p}\times \vec{E}\] and it will tend to rotate.
2. A similar statement can be made about magnetic dipole in a magentic field.
3. A related example is that of spin in a uniform magnetic field. The Hamitonian is of the form \(H=-\vec{\mu}\cdot\vec{B}\) and the equation of motion takes the form \[ i\hbar\dd[\vec{S}]{t} = [\vec{S},H]_- \] With magnetic moment being proportional to the spin \(\vec{S}\), the right hand side becomes proportional to \(\vec{B}\times \vec{S}\).