Tangent and Normal

To a function f(x,y,z)=0 at $(x_0,y_0,z_0)$ #

Tangent #

$$ (x-x_0)\left(\frac{\partial f}{\partial x}_{(x_0,y_0,z_0)}\right) + (y-y_0)\left(\frac{\partial f}{\partial y}_{(x_0,y_0,z_0)}\right) + (z-z_0)\left(\frac{\partial f}{\partial z}_{(x_0,y_0,z_0)}\right)=0 $$

$$\Leftrightarrow (\vec{r}-\vec{r_0})\cdot \nabla f(\vec{r})_{\vec{r_0}} = 0$$

Normal #

$$ \frac{(x-x_0)}{\left(\frac{\partial f}{\partial x}_{(x_0,y_0,z_0)}\right)}=\frac{(y-y_0)}{\left(\frac{\partial f}{\partial y}_{(x_0,y_0,z_0)}\right)}=\frac{(z-z_0)}{\left(\frac{\partial f}{\partial z}_{(x_0,y_0,z_0)}\right)} $$

Normal Vector #

$$ (x_0,y_0,z_0)+t\cdot \left<\left(\frac{\partial f}{\partial x}_{(x_0,y_0,z_0)}\right),\left(\frac{\partial f}{\partial y}_{(x_0,y_0,z_0)}\right),\left(\frac{\partial f}{\partial z}_{(x_0,y_0,z_0)}\right)\right> ⇔ \vec{r_0}+t\cdot \nabla f(\vec{r})_{\vec{r_0}} $$