User:
• Matrices
• Algebra
• Geometry
• Graphs and functions
• Trigonometry
• Coordinate geometry
• Combinatorics
 Suma y resta Producto por escalar Producto Inversa
 Monomials Polynomials Special products Equations Quadratic equations Radical expressions Systems of equations Sequences and series Inner product Exponential equations Matrices Determinants Inverse of a matrix Logarithmic equations Systems of 3 variables equations
 2-D Shapes Areas Pythagorean Theorem Distances
 Graphs Definition of slope Positive or negative slope Determine slope of a line Equation of a line Equation of a line (from graph) Quadratic function Parallel, coincident and intersecting lines Asymptotes Limits Distances Continuity and discontinuities
 Sine Cosine Tangent Cosecant Secant Cotangent Trigonometric identities Law of cosines Law of sines
 Equations of a straight line Parallel, coincident and intersecting lines Distances Angles in space Inner product

Planes in the Space

Intersection of Two Planes

Two arbitrary planes may be parallel, intersect or coincide:

 Parallel Planes Intersecting Planes Coincident Planes
• Parallel planes: Parallel planes are planes that never cross. The ceiling of a room (assuming it’s flat) and the floor are parallel planes (though true planes extend forever in all directions).

• Intersecting planes: Intersecting planes are planes that cross, or intersect. When planes intersect, the place where they cross forms a line. The floor and a wall of a room are intersecting planes, and where the floor meets the wall is the line of intersection of the two planes.

• Coincident planes: Two planes are coincident when they are the same plane. In general, two planes are coincident if the equation of one can be rearranged to be a multiple of the equation of the other

How to find the relationship between two planes.

Given two planes:

$\pi_1\;\equiv\;A_1x+B_1y+C_1z+D_1=0\;$

$\pi_2\;\equiv\;A_2x+B_2y+C_2z+D_2=0\;$

Form a system with the equations of the planes and calculate the ranks.

$\left{A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0$

r = rank of the coefficient matrix
r'= rank of the augmented matrix

The relationship between the two planes can be described as follow:

 Position r r' Intersecting 2 2 $\frac{A_1}{A_2}\;\neq\;\frac{B_1}{B_2}\;\neq\;\frac{C_1}{C_2}$ Parallel 1 2 $\frac{A_1}{A_2}\;=\;\frac{B_1}{B_2}\;=\;\frac{C_1}{C_2}\;\neq\;\frac{D_1}{D_2}$ Coincident 1 1 $\frac{A_1}{A_2}\;=\;\frac{B_1}{B_2}\;=\;\frac{C_1}{C_2}\;=\;\frac{D_1}{D_2}$

State the relationship between the planes:

$\left{x+y-5z+4\;=\;0\\3x-y+15z-1\;=\;0$

 $M_1=\left(\begin{matrix}1&1&-5\\3&-1&15\\nd{matrix}\right)$ $M_2=\left(\begin{matrix}1&1&-5&-4\\3&-1&15&1\\nd{matrix}\right)$

 1 1 3 -1
$\neq\;0$

Therefore r=2 and r'=2. They are Intersecting Planes.

State the relationship between the planes:

$\left{\;x+y-5z+4\;=\;0\\-3x-3y+15z-1\;=\;0$

$\frac{1}{-3}\;=\;\frac{1}{-3}\;=\;\frac{-5}{15}\;\neq\;\frac{4}{-1}$

Parallel Planes

State the relationship between the planes:

$\left{x+y-5z+4\;=\;0\\-3x-3y+15z-12\;=\;0$

$\frac{1}{-3}\;=\;\frac{1}{-3}\;=\;\frac{-5}{15}\;\neq\;\frac{4}{-12}$

Coincident Planes

 Planes = 8x+2y-6z+7=0 and = -4x+9y+2z+9=0 are: Parallel Coincident Intersecting