Find $\mathbf{T}, \mathbf{B},$ and $\mathbf{N}$ at the given point:

$\mathbf{r}(t) = \langle \ln \cos t, \sin t, \tan^2 t \rangle$ at $ (1,0,0)$.

Find the equations of the normal plane and osculating plane of the curve at the given point

$x = t, y = t^2, z = ^3, (1,1,1)$.

Vector Relationships

Show that $d\mathbf{B}/ds$ is perpendicular to $\mathbf{B}$

Show that $d\mathbf{B}/ds$ is perpendicular to $\mathbf{T}$

Use the previous two relationships to show that $d\mathbf{B}/ds = - \tau(s) \mathbf{N}$ for some number $\tau(s)$ called the torsion of the curve.

Show that for a plane curve the torsion is $\tau(s) = 0$.