If you let then will be normal to the level surface . Calculate and find where it's parallel to the normal of the given plane.
We're doing gradient vectors, and I'm kind of confused by the applications of them.
Here's a question where I'm not entirely sure where to start:
Find the points on the ellipsoid where the tangent plane is parallel to the plane
First, I know that the normal plane to the ellipsoid at these points is going to be <3,-1,3>. I know that to find the gradient vector I take the partials with respect to x and y, but then I'm not sure where to go from there. I don't have points I'm working with, I'm being asked to find points. The whole directional derivative thing is actually very confusing in general--if someone could help me out that would be great!
Okay, I did that (using n for lambda here) and got that and by plugging in I got that . Then I calculated the points to be and . Is that correct?
Thank you for helping me. If you don't mind me asking, could you explain the logic behind the method used to solve this problem? I'm trying to fully understand it, but I can't figure out how it came down to in order to find the solution.
Your value looks incorrect but you have the right method.
The logic behind this is that is normal to the surface at the point (well known property of ). Hence it's normal to the tangent plane to the surface at that point. We're simply looking for points where this normal is parallel to the normal of the given plane. Two vectors are parallel if one is a scalar multiple of the other.