Lecture 54: Planes and Lines in Space

54.1 Plane and Line Equations

54.1.1 Plane Equations

#####Definition: Planeequations

description:

$1\text{ ) General formula: }\quad Ax+By+Cz+D=0.\quad\mathbf{n}=\{A,B,C\}$

$2\text{ ) Point French: }\quad A(x-x_0)+B(y-y_0)+C(z-z_0)=0$

$3\text{ ) Intercept type: }\quad\frac xa+\frac yb+\frac zc=1$

54.1.2 Line Equations

#####Definition: Alinearequation

description:

$$1)\text{ General formula: }\quad\begin{cases}A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\end{cases}$$

$$2)\text{ Symmetric Formula: }\quad\frac{x-x_0}l=\frac{y-y_0}m=\frac{z-z_0}n$$

$$3)\text{ Parametric expression }:\,x=x_0+lt,y=y_0+mt,z=z_0+nt.$$

54.2 Distance Relations

54.2.1 Positional Relationship Between Plane and Straight Line

Core: Plane -> Normal Vector; Straight Line -> Direction Vector

54.2.2 Distance from Point to Surface

Concept: The distance from point $(x_0,y_0,z_0)$ to plane $Ax+By+Cy+D=0$

• $$d=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$$

54.2.3 Distance from Point to Straight Line

Concept: $\text{ Point }({x_0,y_0,z_0})\text{ To a straight line }\frac{x-x_1}l=\frac{y-y_1}m=\frac{z-z_1}n$

• $$d=\frac{|\{x_1-x_0,y_1-y_0,z_1-z_0\}\times\{l,m,n\}|}{\sqrt{l^2+m^2+n^2}\quad J}$$