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I am given a parametric surface $S,$ defined by $\mathbf{r} (u, v) = (x(u, v), y(u, v), z(u, v)),$ and a vector $\mathbf{a} = (a_{1}, a_{2}, a_{3}).$ I need to detrmine if the vector is tangent or normal to the surface.

Now, what did I try to do? To see if the vector is tangent to $S$ I calculated the normal vector $\mathbf{N} = \partial_{u} r \times \partial_{v} r$ and calculated the dot product between $\mathbf{a}$ and $\mathbf{N}$ to see if it's zero. On the other hand, to see if the vector is normal to $S$ I tried to see if it was a multiple of $\mathbf{N}.$

The thing is that I am not getting any affirmative answer by these methods, so I fear I am doing something wrong. So I would like to ask what it could be.

Lucas Romero
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  • You are doing well, theoretically. Just pay attention to your computations, as it is easy to make errors there. Note carefully that there is no dichotomy between "tangent" and "normal" vectors; in other words, there exist vectors that are neither tangent nor normal to a given surface. For a simpler example, consider the flat surface in $\mathbb{R}^3$ $$S:\ z=0, $$ and the vector $\vec{v}=(1,1,1)$. – Giuseppe Negro May 18 '14 at 21:14
  • You're right. However, to see if it's normal to S, try to multiply $a$ by $\partial_{u} r$ and $\partial_{v} r$ and check if the product is 0. – Imanol Pérez Arribas May 18 '14 at 21:15

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