Finding Tangent Line to Graph

Question

27. Find a vector equation for the tangent line to the curve of intersection of the cylinders

    $\ x^2 + y^2 = 25$ and $\ y^2 + z^2 = 25$

    at the point (3,4,2).

I don't understand the answer key. I've explained my interpretations below. Could someone clarify:

Why is the projection of C onto the xy plane:$\ x^2 + y^2 = 25$. I understand that a projection in 3D space is the "shadow" of the graph onto one plane, but I don't quite see that C's projection can be modeled as given.

Why is z >= 0?

Answer 1

The intersection is a subset (or, if you prefer, contained within) the cylinder given by $x^2+y^2=25$, right? That equation is not sufficient to determine the points on the curve, but they all do fulfill it, because one thing the intersection is is exactly being part of that cylinder.

Now, if you take any point on the cylinder and project it onto the $x/y$ plane, what do you get? You get a point with $z=0$ and $x^2+y^2=25$. Try with the given point, $3^2+4^2=25$.

As for $z>0$, you are in the vicinity of $(3,4,2)$, and for that point, $z=2>0$.

Answer 2

Why is the projection of C onto the xy plane: $x2+y2=25$.

You are dealing with a cylinder that is not dependent on z. So imagine with edges 5 units away from the z-axis in all directions and that is your cylinder. Or in other words, the cylinder defined by $x^2+y^2=25$ has the z axis running through the middle of the cylinder (think of it like a tube surrounding the z-axis). Projecting this onto the xy-plane would just be a circle of radius 5. You can think of projecting onto the xy-plane as looking down the z-axis. If you look from a cylinder from the top, you see a circle. Looking down the z-axis is the same as looking at the top of this cylinder (even though it's more of an infinite tube, I think you can get the idea).

Answer 3

The projection of the cylinder $x^2 + y^2 = 25$ onto the $xy$ plane is the circle $x^2 + y^2 = 25,\,z = 0$. Any intersection with the cylinder will lie on the cylinder. Thus the projection of the intersection will be some part of the projection of the cylinder.

$z$ is close to 2 near the point $(3,\,4,\,2)$ and is thus more than 0.