Planes in Three Dimensions

Plane forms

Point direction form:

\( a(x-x_1) + b(y-y_1) + c(z-z_1) = 0 \)

where \(P(x_1, y_1, z_1)\) lies in the plane, and the direction (a,b,c) is normal to the plane.

General form:

\( Ax + By + Cz + D = 0 \)

where direction (A,B,C) is normal to the plane.

Intercept form:

\( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \)

this plane passes through the points (a,0,0),(0,b,0) and (0,0,c).

Three point form:

\(
\begin{vmatrix} x-x_3 & y-y_3 & z-z_3 \\ x_1-x_3 & y_1-y_3 & z_1-z_3 \\ x_2-x_3 & y_2-y_3 & z_2-z_3 \end{vmatrix} = 0
\)

Normal form:

\( x\,\cos \alpha + y\,\cos\beta + z\,\cos\gamma = p \)

Parametric form:

\(
\begin{aligned}
x &= x_1 + a_1\,s + a_2\,t \\
y &= y_1 + b_1\,s + b_2\,t \\
z &= z_1 + c_1\,s + c_2\,t \end{aligned}
\)

where the directions \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) are parallel to the plane.

Angle between two planes:

The angle between planes \(A_1x + B_1y + C_1z + D_1 = 0\) and \(A_2x + B_2y + C_2z + D_2 = 0\) is:

\( \alpha = \arccos \frac{A_1A_2 + B_1B_2 + C_1C_2} {\sqrt{A_1^2 + B_1^2 + C_1^2} \cdot \sqrt{A_2^2 + B_2^2 + C_2^2}} \)

The planes are parallel if and only if

\( \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \)

Equation of a plane

The equation of a plane through \(P_1(x_1, y_1, z_1)\) parallel to directions \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) has an equation:

\( \begin{vmatrix} x-x_1 & y – y_1 & z – z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \)

The equation of a plane through \(P_1(x_1, y_1, z_1)\) and \(P_1(x_2, y_2, z_2)\) are parallel to directions \((a_1, b_1, c_1)\) and \((a, b, c)\) has an equation:

\( \begin{vmatrix} x-x_1 & y – y_1 & z – z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a & b & c \end{vmatrix} = 0 \)

The equation of a plane through \(P_1(x_1, y_1, z_1)\) , \(P_2(x_2, y_2, z_2)\) and \(P_3(x_3, y_3, z_3)\) , has equation\

\(
\begin{vmatrix} x-x_1 & y – y_1 & z – z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0
\)

Distance from point to plane

The distance of \(P_1(x_1, y_1, z_1)\) from the plane \(Ax + By + Cz + D = 0\) is

\( d = \frac{Ax_1 + By_1 + Cz_1}{\sqrt{A^2 + B^2 + C^2}} \)

Intersection of two planes

The intersection of planes \(A_1x + B_1y + C_1z + D_1 = 0\) and \(A_2x + B_2y + C_2z + D_2 = 0\) is the line:

\( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \)

where

\(
\begin{aligned}
a &= \begin{vmatrix} B_1 & C_1 \\ B_2 & C_2 \end{vmatrix}~~
b = \begin{vmatrix} C_1 & A_1 \\ C_2 & A_2 \end{vmatrix}~~
c = \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix} \\
x_1&= \frac{b\begin{vmatrix}D_1& C_1 \\ D_2 & C_2 \end{vmatrix} –
c\begin{vmatrix}D_1& B_1 \\ D_2 & B_2 \end{vmatrix} }{a^2 + b^2 + c^2} \\
y_1&= \frac{c\begin{vmatrix}D_1& A_1 \\ D_2 & A_2 \end{vmatrix} –
a\begin{vmatrix}D_1& C_1 \\ D_2 & C_2 \end{vmatrix} }{a^2 + b^2 + c^2} \\
z_1&= \frac{a\begin{vmatrix}D_1& B_1 \\ D_2 & B_2 \end{vmatrix} –
b\begin{vmatrix}D_1& A_1 \\ D_2 & A_2 \end{vmatrix} }{a^2 + b^2 + c^2}
\end{aligned}
\)

If a=b=c=0, then the planes are parallel.

 

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