Originally Posted by

**bkarpuz** Dear **MHF** members,

I have the following problem.

**Problem.** Let the curve (ellipse) $\displaystyle \alpha$ be the intersection of the surfaces $\displaystyle x^{2}+2y^{2}+4z^{2}=10$ and $\displaystyle x-2y=0$. Find the tangent vector of $\displaystyle \alpha$ at the point $\displaystyle (2,1,-1)$.

I was just trying to find the direction of the tangent vector. I tried to find it by using the cross product of the gradients of the surfaces at the given point (as this vector should be in the same direction with the tangent vector), and get the result $\displaystyle (-16,-8,-12)$. But when I try to solve it by using the derivative of the parametric form of the curve at the given point, I get a different result, which is $\displaystyle (2\sqrt{2/3},\sqrt{2/3},\sqrt{3/2})$. Up to me, both of these results should be the same. Please help me.

Thanks a lot.

**bkarpuz**