(Solved): The space curve \( C \) is given by the vector function \[ \mathbf{r}(t)=\langle\cos 2 t, \sin 2 t, ...

The space curve \( C \) is given by the vector function \[ \mathbf{r}(t)=\langle\cos 2 t, \sin 2 t, t\rangle . \] Consider the point \( P=\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}, \frac{\pi}{3}\right) \) on the curve. 1.) Find the unit tangent vector \( \mathbf{T} \) and the unit normal vector \( \mathbf{N} \) at the point \( P \). Show that the vectors you found in problem 2 are orthogonal. (Note that the unit tangent vector and unit normal vector will always be orthogonal, so if they are not then check you work - there is a mistake somewhere.) a) Now find the binormal vector \( \mathbf{B} \) at the point \( P \). Show that the binormal vector you found in problem 4 is a unit vector. (Note that the binormal vector will always be unit length, so if it is not then check your work.) b.) Find equation for the osculating plane for \( C \) at the point \( P \). Write your answer in the form \( a x+b y+c z+d=0 \), but note that it will be impossible to make \( a, b, c \), and \( d \) all integers, so don't even try.