Mastering Planes, Quadratic Surfaces, and 3D Coordinates

Planes in 3D Space

Equations for planes parallel to coordinate planes:

• x = a (parallel to yz-plane)
• y = b (parallel to xz-plane)
• z = c (parallel to xy-plane)

Example: Plane Through a Point

Find the equation of the plane passing through (3, -1, 7) and perpendicular to the vector n = <4, 2, -3>:

4(x - 3) + 2(y + 1) - 3(z - 7) = 0

Example: Plane Through Three Points

Find the equation of the plane through P₁(1, 2, -1), P₂(2, 3, 1), and P₃(3, -1, 2):

P₁P₂ = <1, 1, 2>; P₁P₃ = <2, -3, 3>

n = P₁P₂ × P₁P₃ = <9, 1, -5>

Equation: 9(x - 1) + 1(y - 2) - 5(z + 1) = 0

Example: Line and Plane Parallelism

Determine if the line x = 3 + 8t, y = 4 + 5t, z = -3 - t is parallel to the plane x - 3y + 5z = 12.

The line is parallel to the plane if it is perpendicular to the normal vector n = <1, -3, 5>.

n · V = (1)(8) + (-3)(5) + (5)(-1) = 8 - 15 - 5 = -12 ≠ 0 (Not parallel).

Example: Angle Between Two Planes

Formula: cos θ = |n₁ · n₂| / (||n₁|| ||n₂||)

For 2x - 4y + 4z + 6 = 0 and 6x + 2y - 3z = 4:

n₁ = <2, -4, 4>, n₂ = <6, 2, -3>

n₁ · n₂ = (2)(6) + (-4)(2) + (4)(-3) = -8

cos θ = |-8| / (6 * 7) = 4/21; θ = cos^-1(4/21) ≈ 79°

Distance Problem Involving Planes

D = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)

Example: Distance from Point to Plane

Find the distance between (1, -4, -3) and 2x - 3y + 6z + 1 = 0:

D = |2(1) - 3(-4) + 6(-3) + 1| / √(2² + (-3)² + 6²) = 3/7

Quadratic Surfaces

• Ellipsoid: x²/a² + y²/b² + z²/c² = 1
• Elliptic Cone: z² = x²/a² + y²/b²
• Hyperboloid of One Sheet: x²/a² + y²/b² - z²/c² = 1
• Elliptic Paraboloid: z = x²/a² + y²/b²
• Hyperboloid of Two Sheets: z²/c² - x²/a² - y²/b² = 1
• Hyperbolic Paraboloid: z = y²/b² - x²/a²

Cylindrical and Spherical Coordinates

Cylindrical to Rectangular

x = r cos θ, y = r sin θ, z = z

Rectangular to Cylindrical

r = √(x² + y²), tan θ = y/x, z = z

Spherical to Rectangular

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ

Rectangular to Spherical

ρ = √(x² + y² + z²), tan θ = y/x, cos φ = z/ρ