Key takeaways:

• Equation of plane, normal vector of a plane, how to find a plane knowing 3 points.

• Angle between two planes

• Distance of a point to a plane

1 Equation of plane

If $\mathbf{\text{n}}=(a,b,c)$ is a given vector, ${\mathbf{\text{r}}}_0=(x_0,y_0,z_0)$ is the position vector for $P_0(x_0,y_0,z_0)$, then the plane going through $(P)$ going through $P_0$ and is perpendicular to $\mathbf{\text{n}}$ is given by

• Vector form $\mathbf{\text{r}}=(x,y,z)\in (P)$ if

  $$\mathbf{n}\cdot (\mathbf{r}-{\mathbf{r}}_0)=0$$

• Parametric form $\mathbf{\text{r}}=(x,y,z)\in (P)$ if

  $$\mathbf{n}\cdot (\mathbf{r}-{\mathbf{r}}_0)=0⟹a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

• Other parametric form sometime we write

  $$ax+by+cz-\underset{d}{\underbrace{(a{x}_0+b{y}_0+c{z}_0)}}=0,\text{or}ax+by+cz=d.$$

2 Distance from a point to a plane