Question

Prove that the graph of the vector function $\mathbf{r}\left(t\right)=⟨t\sin t,t\cos t,t⟩,\text{where}t\ge 0$, is a conical helix by showing that it lies on the graph of the cone described by $z=\sqrt{x^2+y^2}$

Short answer

Ans: It is proved that vector function $\mathbf{r}\left(t\right)=⟨t\sin t,t\cos t,t⟩,\text{where}t\ge 0$is a conical helix by showing that it lies on the graph of the cone described by $z=\sqrt{x^2+y^2}$

Step-by-step solution

Step 1. Given information.

given,

$\mathbf{r}\left(t\right)=⟨t\sin t,t\cos t,t⟩,\text{where}t\ge 0$

Step 2. The objective is to show that the graph of a vector function r(t)=⟨tsin⁡t,tcos⁡t,t⟩ is a conical helix and lies on the graph z=x2+y2.

Substitute $x=t\sin t,y=t\cos t\text{and}z=t\text{in}z=\sqrt{x^2+y^2}\text{.}$

$\begin{array}{r}t=\sqrt{t^2{\sin }^{2}t+t^2{\cos }^{2}t}\\ t=\sqrt{t^2\left({\sin }^{2}t+{\cos }^{2}t\right)}\\ t=\sqrt{t^2}\\ t=t\end{array}$

This shows that role="math" localid="1649614921747" $\mathbf{r}\left(t\right)=⟨t\sin t,t\cos t,t⟩$and role="math" localid="1649614828344" $z=\sqrt{x^2+y^2}$represents the same curve in the parametric and Cartesian systems respectively.

Step 3. Now,

Graph of $z=\sqrt{x^2+y^2}$

Step 4. And

Graph of$\mathbf{r}\left(t\right)=⟨t\sin t,t\cos t,t⟩$