# Tangent/Normal planes, intersections, vector tangent

## Homework Statement

Surface (s) given by

x^3*+5*y^2*z-z^2+x*y=0

question ask to find normal and tangent plane to S at the point P=(1,-1,0) then find vector tangential to curve of intersection of S and the plane x+z=1 at point P.

## The Attempt at a Solution

So I started off finding the normal first using n=(fx,fy,-1) which gave me n=(-2,-1,1).

Then proceeding to the tangent plane using the tangent plane equation fx(x-x0)+fy(y-y0)-(z-z0)=0, which gave me the tangent plane 2*x+y-z-1=0.

Please correct me if my method is right or wrong for the above.

But my real question is the last part of the problem statement, which is to find the vector tangential to the curve intersecting x+z=1.

My guess is to find the gradient of the tangent vector and the gradient of the given curve(x+z=1) and take the cross product of the 2? I have no idea where to begin the last, part, any hint or help would be much appreciated.

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Surface (s) given by

x^3*+5*y^2*z-z^2+x*y=0

question ask to find normal and tangent plane to S at the point P=(1,-1,0) then find vector tangential to curve of intersection of S and the plane x+z=1 at point P.

## The Attempt at a Solution

So I started off finding the normal first using n=(fx,fy,-1) which gave me n=(-2,-1,1).

That formula for the normal is for when the equation is in the form z = f(x,y). Your equation is not in that form but in the implicit form f(x,y,z)=0. Use $\nabla f$.

Then proceeding to the tangent plane using the tangent plane equation fx(x-x0)+fy(y-y0)-(z-z0)=0, which gave me the tangent plane 2*x+y-z-1=0.

Please correct me if my method is right or wrong for the above.

But my real question is the last part of the problem statement, which is to find the vector tangential to the curve intersecting x+z=1.

My guess is to find the gradient of the tangent vector and the gradient of the given curve(x+z=1) and take the cross product of the 2? I have no idea where to begin the last, part, any hint or help would be much appreciated.

The "gradient of the tangent vector" doesn't make any sense. Just cross the normals to the two surfaces.