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 3: Line Equations

This page details various representations of lines in three-dimensional space.

Definition: A line in space can be defined by:

• A point P₀(x₀, y₀, z₀)
• A direction vector v(l,m,n)

Example: Parametric equations of a line: x = x₀ + lt y = y₀ + mt z = z₀ + nt

Highlight: The line passing through two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) has direction vector: v = $x₂-x₁, y₂-y₁, z₂-z₁$

Vocabulary: Cartesian equations of a line are obtained by eliminating the parameter t from parametric equations when l,m,n ≠ 0: $x-x₀$/l = $y-y₀$/m = $z-z₀$/n

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."

Relative Positions of Lines and Planes

This section discusses the relationships between lines and planes in space, including rette parallele and rette perpendicolari.

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

• Parallel if v = kv' for some k≠0
• Perpendicular if ll' + mm' + nn' = 0

Highlight: Lines in space can be:

• Parallel
• Intersecting
• Skew (neither parallel nor intersecting)

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)

Line Representations in Space

This section examines different ways to represent lines in three-dimensional space.

Definition: A line can be represented in:

• Explicit form: y = mx + q
• Implicit form: ax + by + c = 0
• Parametric form: P₀ + tv

Highlight: Each representation has its advantages:

• Explicit form gives clear geometric meaning
• Implicit form includes vertical lines
• Parametric form provides direction information

Page 1: Coordinates and Vectors in Space

This page introduces fundamental concepts of three-dimensional coordinate systems and vector operations.

Definition: A three-dimensional coordinate system consists of three perpendicular axes (x, y, z) intersecting at the origin O.

Vocabulary: A point P in space is represented by an ordered triple (xp, yp, zp), where:

• xp is the abscissa
• yp is the ordinate
• zp is the height/elevation

Example: The distance between two points A(xa, ya, za) and B(xb, yb, zb) is calculated using: d = √[(xb-xa)² + (yb-ya)² + (zb-za)²]

Highlight: Vector operations in space include:

• Addition: a + b = (ax+bx, ay+by, az+bz)
• Scalar multiplication: ka = (kax, kay, kaz)
• Dot product: a·b = axbx + ayby + azbz
• Cross product: axb = (aybz-azby)i + (azbx-axbz)j + (axby-aybx)k