# Angle Between two surfaces and a point

1. Sep 2, 2008

### CompStang

Find the angle between the surfaces defined by r^2= 9 and x + y + z^2= 1 at the point (2,-2,1)? --- I know this should be extremely simple but it is blowing my mind for some reason. Any help would be greatly appreciated.

2. Sep 3, 2008

### HallsofIvy

Staff Emeritus
The angle between two surfaces at a point pf intersection is the angle between their normal vectors at that point. And if you write the surfaces as f(x,y,z)= constant, the normal vector is in the same direction as $\nabla f$. You probably want to write the first surface in Cartesian coordinates as x2+ y2+ z2= 9. Find the gradients of those two functions and then use the fact that
$\vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta)$ where $\theta$ is the angle between the two vectors.

Although this was posted in Precalculus, I don't believe this can be done without using Calculus.

Last edited: Sep 3, 2008