Planes

Planes

Definitions

Geometric

$$\mathcal{P}=\{P\mid AB\perp AP\}$$

Algebraic

• Values
  • Point on plane: $P_0=(x_0,y_0,z_0)$
  • Normal vector: $\vec{n}=⟨n_1,n_2,n_3⟩$
• Equation
  • $n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0$
  • $n_{1}x+n_{2}y+n_{3}z=\vec{n}\cdot \overrightarrow{OP}=d$
• Important Bits
  • $P\in \mathcal{P}⟺\vec{n}\perp \overrightarrow{P_{0}P}⟺\vec{n}\cdot \overrightarrow{P_{0}P}=0$
  • Equation: $n_{1}x+n_{2}y+n_{2}z=d$

Properties

• Normal vector: $\vec{n}\perp \mathcal{P}$
  • Can be used to find if other vectors are parallel/perpendicular
  • Can be used to find the angle between planes
• A particular point: $P_0\in \mathcal{P}$

Coordinate Planes

• Planes that are only offset in a single axis
• In the form $x/y/z=c$

Parallel Planes

• Planes are parallel if their normal vectors are the same
  • so only the d value is different in the algebraic definition
• Distance between parallel planes

$$D=\frac{|d_1-d_2|}{\Vert \vec{n}\Vert }$$

Points to Plane

$$\begin{gathered} \vec{n}=\overrightarrow{AB}\times \overrightarrow{AC} \\ {P}_0=A \end{gathered}$$

Distance From a Point to a Plane

$$D=\left|\frac{\overrightarrow{P_{0}P}\cdot \vec{n}}{\Vert \vec{n}\Vert }\right|$$