Editing Rotation (section)

=== Rotation of vectors ===
A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector <math> \vec A </math> which is parameterized by some variable <math display="inline"> t </math> for which:

<math>{d|\vec A|^2 \over dt}={d(\vec A\cdot \vec A) \over dt} \Rightarrow   {d|\vec A| \over dt}={d\vec A  \over dt}\cdot \hat{A}</math>

Which also gives a relation of rate of change of unit vector by taking <math> \vec A </math>, to be such a vector: <math display="block"> {d\hat A \over dt}\cdot \hat A = 0 </math>showing that <math display="inline"> {d\hat A \over dt} </math> vector is perpendicular to the vector, <math> \vec A </math>.<ref>{{Cite book |last1=Kumar |first1=N. |title=Generalized motion of rigid body |last2=Kumar |first2=Naveen |date=2004 |publisher=Alpha Science International Ltd |isbn=978-1-84265-160-5 |location=Pangbourne, U.K. |pages=5}}</ref>

From:

<math>{d\vec A \over dt} = {d(|\vec A|\hat A) \over dt} = {d|\vec A| \over dt}\hat{A}+|\vec A|\left({d\hat A \over dt}\right)
</math>,

since the first term is parallel to <math> \vec A </math> and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against the vector itself.