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 \(\textbf{n} = (a,b,c)\) is a given vector, \(\textbf{r}_0 = (x_0,y_0,z_0)\) 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 \(\textbf{n}\) is given by

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

  \begin{equation*} \mathbf{n}\cdot (\mathbf{r} - \mathbf{r}_0) = 0 \end{equation*}

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

  \begin{equation*} \mathbf{n}\cdot (\mathbf{r} - \mathbf{r}_0) = 0 \qquad \Longrightarrow \qquad a(x-x_0) + b(y-y_0)+c(z-z_0) = 0 \end{equation*}

• Other parametric form sometime we write

  \begin{equation*} ax+by+cz - \underbrace{(ax_0+by_0+cz_0)}_{d} = 0, \qquad \text{or}\qquad ax+by+cz = d. \end{equation*}

2 Distance from a point to a plane