User:
• Matrices
• Algebra
• Geometry
• Funciones
• 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 Ecuación de una recta Equation of a line (from graph) Quadratic function Posición relativa de dos rectas Asymptotes Limits Distancias Continuity and discontinuities
 Sine Cosine Tangent Cosecant Secant Cotangent Trigonometric identities Law of cosines Law of sines
 Ecuación de una recta Posición relativa de dos rectas Distancias Angles in space Inner product

Angles

# Plane-Plane Angle

Two distinct planes in three-space either are parallel or intersecting in a line. If they intersect, you can determine the angle $(0\;\le\;\theta\;\le\;\frac{\pi}{2})\;$ between them from the angle between their normal vectors.

Specifically, if vectors n1=(A1,B1,C1) and n2=(A2,B2,C2) are normal to two intersecting planes,

$\pi_1\;\equiv\;A_1x+B_1y+C_1z+D_1=0\;$ and $\pi_2\;\equiv\;A_2x+B_2y+C_2z+D_2=0\;$

the angle θ between the normal vectors is equal to the angle between the two planes and is given by:

 That is:

Consequently, two planes with normal vectors n1 and n2 are

1. perpendicular if n1·n2=0
2. parallel if n2=kn1, for some nonzero scalar k
 Perpendicular planes Parallel planes

Determine the angle between the planes

$\fs2\;\pi_1\;\equiv\;2x-y+z-1=0\;and\;\pi_2\;\equiv\;x+z+3\;=\;0$

Solution:

n1=(2,-1,1) and n2=(1,0,1)

θ=30º