Page 2: General Equation of a Plane

This page covers the fundamental concepts of planes in three-dimensional space.

Definition: The general equation of a plane through point P(x₀, y₀, z₀) with normal vector (a,b,c) is: a(x-x₀) + b(y-y₀) + c(z-z₀) = 0

Example: Special cases of planes:

• When c=0: plane parallel to z-axis
• When b=0: plane parallel to y-axis
• When a=0: plane parallel to x-axis

Highlight: Two planes are parallel if and only if their normal vectors are parallel: a/a' = b/b' = c/c'

Quote: "The distance from a point A(xa, ya, za) to a plane ax + by + cz + d = 0 is given by: |axa + bya + cza + d| / √(a² + b² + c²)"

Page 4: Relative Positions of Lines and Planes

This page explores the relationships between lines and planes in three-dimensional space.

Definition: Two lines with direction vectors v(l,m,n) and v'(l',m',n') can be:

• Parallel
• Coincident
• Skew
• Intersecting
• Perpendicular

Example: Lines are perpendicular when: ll' + mm' + nn' = 0

Highlight: A line and plane are:

• Parallel when n·v = 0
• Intersecting when n·v ≠ 0
• Perpendicular when n = kv

Quote: "Two perpendicular lines can be either intersecting or skew."

Vector Operations in Space

This section covers vector operations and their geometric interpretations in three-dimensional space.

Definition: A vector in space can be represented as v = (x,y,z) with magnitude |v| = √(x² + y² + z²)

Example: Vector operations include:

• Addition: (x₁,y₁,z₁) + (x₂,y₂,z₂) = (x₁+x₂, y₁+y₂, z₁+z₂)
• Scalar multiplication: k(x,y,z) = (kx,ky,kz)