(Solved): Consider the following curve. x=sin(3t),y=cos(3t),z=12t Using the given parametric equations, gi ...

Consider the following curve. $x=\sin (3t),y=-\cos (3t),z=12t$ Using the given parametric equations, give the corresponding vector equation $\mathbf{r}(t)$. $\mathbf{r}(t)=⟨\sin (3t),-\cos (3t),12t⟩$ Find ${\mathbf{r}}^{\prime }(t)$ and $\left|{\mathbf{r}}^{\prime }(t)\right|$. $\begin{array}{rl}{\mathbf{r}}^{\prime }(t)& =3\cos (3t)\mathbf{i}+3\sin (3t)\mathbf{j}+12\mathbf{k}\\ \left|{\mathbf{r}}^{\prime }(t)\right|& =\sqrt{(3\cos (3t){)}^2+(3\sin (3t){)}^2+144}\end{array}$ Find the equation of the normal plane of the given curve at the point $(0,1,4\pi )$. Now consider the osculating plane of the given curve at the point $(0,1,4\pi )$. Determine each of the following. $\mathbf{T}(t)=⟨\frac{\sin (3t)}{\sqrt{153}},\frac{-\cos (3t)}{\sqrt{153}},\frac{12t}{\sqrt{153}}⟩$ ${\mathbf{T}}^{\prime }(t)=$ $\left|{\mathbf{T}}^{\prime }(t)\right|=$ $\mathbf{N}(t)=$ Find the equation of the osculating plane of the given curve at the point $(0,1,4\pi )$.