T' is perpendicular to T.
Show \(\displaystyle \L\\T(t)\cdot{T'(t)}=0\)
\(\displaystyle \L\\\frac{d}{dt}[T(t)\cdot{T'(t)}]=T(t)\cdot{\frac{dT}{dt}}+\frac{dT}{dt}\cdot{T(t)}\)
\(\displaystyle \L\\\frac{d}{dt}[||T(t)||^{2}]=2T(t)\cdot{\frac{dT}{dt}}\)
But, \(\displaystyle \L\\||T(t)||^{2}\) is a constant, so the derivative is 0.
Therefore, hence, and thus:
\(\displaystyle \L\\2T(t)\cdot{\frac{dT}{dt}}=0\)
