tangent line problem

[renegade05]

The plane \(\displaystyle y+z=3\) intersects the cylinder \(\displaystyle x^2+y^2=5\) in an ellipse. Find parametric equations for the tangent line to this ellipse at \(\displaystyle (1,2,1)\).

I found both normals to be: \(\displaystyle <0,1,1>\) and \(\displaystyle <2,4,0>\) at the point \(\displaystyle (1,2,1)\).

I then took the cross product which is \(\displaystyle <-4,2,-2>\) and got my parametric equation to be :

\(\displaystyle x=1-4s\)
\(\displaystyle y=2+2s\)
\(\displaystyle z=1-2s\)
Where s is parameter.

Am i way off on this one or what? I feel like my logic is incorrect for some reason. Looking for some confirmation/feedback.

 

[galactus]

Yes, you have it, except for an incorrect sign. You should have

\(\displaystyle [4,-2,2]\).

Which gives \(\displaystyle x=1+4t, \;\ y=2-2t, \;\ z=1+2t\)


I normally use t, instead of s, for parameters.

 

[renegade05]

Im not seeing the sign error. Where do you see it?

 

[renegade05]

Yes, you have it, except for an incorrect sign. You should have

\(\displaystyle [4,-2,2]\).

Which gives \(\displaystyle x=1+4t, \;\ y=2-2t, \;\ z=1+2t\)


I normally use t, instead of s, for parameters.

actually Galactus, i dont know why I didnt see this before, but it doesnt matter of the sign.

Since this is just a direction vector for a line. I mean you could plug in any scalar for t and it would still give you a direction vector.

So we are both right