- #1
jegues
- 1,097
- 3
Homework Statement
It is known that the curve C,
[tex]x = 4t -2t^{2} -1, y = 2t^{3} - 1, z=3t^{2} -5[/tex]
and the surface,
[tex]x^{2}y + xz + 1 = 0[/tex]
intersect at the point (1,1,-2). You need NOT show this. Prove that the curve is perpendicular to the surface at the point.
Homework Equations
The Attempt at a Solution
I need to get my ideas straightened out before I can start attacking this one.
We want to show that the curve C is perpendicular to the surface at the point (1,1,-2).
If we stop and think about what we mean by perpendicular to the surface at a point, it means perpendicular to the tangent plane of that surface, at that point.
If we take the Gradient of the surface evaluated at the point we will get a vector that is perpendicular to the tangent plane of the surface at that point.
Now I think what I want to show is that the curve is pointing in the same(or opposite) direction of the Gradient of the surface at that point, thus the curve will also be perpendicular.
I could take the cross product between the Gradient of the surface and the curve, and if is this equals 0 then I know that they are parallel. But then again, I don't have a vector representation of the curve.
I think the part I'm confused about is how to represent my curve C in the form of a vector.
One can note that when t = 1, we obtain the point (1,1,-2) on the curve.
Do I have my ideas straight? Can anyone help me finish this one off?
Thanks again!