Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$.
I go through the partial derivatives to get the tangent vector as $\begin{bmatrix} x\\ y\\ z \end{bmatrix}= \begin{bmatrix} 2x\\ 2y\\ 2z \end{bmatrix}$.
Now, I put the equation together this way $\begin{bmatrix} 2x\\ 2y\\ 2z \end{bmatrix}\cdot \begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}=0$ But I can't solve for the point because there are 3 unknowns and there will be many possible solution to $x,y,z$. Is my tangent vector correct in the first place? What should I do to to solve for the exact point?
Edit
I happen to find the same equation of the sphere in a book. Somehow, it says that the tangent vector of a point on the sphere is $\begin{bmatrix} 0\\ 1\\ \frac{y}{\sqrt{5^2-y^2}} \end{bmatrix}$. But how come it has this tangent vector different from mine. Which is correct? And how was this derived?