Question

for the curve x=3cost,y=3sint,z=4t find the unit tangent vector

Answer 1

differentiate and then divide by magnitude:\[r(t)= \left(\begin{matrix}3cost \\ 3sint \\ 4t\end{matrix}\right)\]

\[v(t) = \left(\begin{matrix} -3sint \\ 3cost \\ 4\end{matrix}\right)\]

\[T(t) = \frac{v(t)}{|v(t)|} = \frac{1}{\sqrt((-3sint)^2 +(3cost)^2 + 4^2)}\left(\begin{matrix} -3sint \\ 3cost \\ 4\end{matrix}\right)\]

which should simplify to:

\[T(t) = \frac{1}{5}\left(\begin{matrix} -3sint \\ 3cost \\ 4\end{matrix}\right)\]