One of the questions in my homework was:
"Show that the curve $\vec{r}(t)=\cos t \vec{i}+\sin t \vec{j}+(1-\cos t)\vec{k}$ is an ellipse by showing that it is the intersection of a cylinder and a plane. Find equations for the cylinder and the plane."
It is easy to see that the curve is the intersection of the cylinder $x^2 + y^2 =1$ and the plane $x+z=1$.
Maybe I'm misunderstanding, but the question makes it sound like the hard part is showing that the curve is the intersection of a cylinder and a plane and that once they are found it is obvious that the curve is an ellipse. I have no idea how I can prove that the curve is an ellipse. Since it is not parallel to the $xy$ plane (or any other "conventional" planes), I am having a hard time showing that it satisfies the equation of an ellipse.
Edit: I should add that I have not taken a linear algebra course yet, so please bear that in mind when posting a solution/hint. (I’m only adding this because many of the multivariable calculus hints online assume I am familiar with linear algebra.)
Best Answer
Sometimes a figure really helps craft the algebra: