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Grade 12 Math Lab: Analysis & Inference FREE
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Planes in 3D Space
Learning Objectives
- Write plane equations
- Find normal vectors
- Calculate distances and intersections
Key Concepts
Plane Equation
ax + by + cz = d where ⟨a,b,c⟩ is the normal vector
Normal Vector
Vector perpendicular to the plane
Theory
**Plane equation:** ax + by + cz = d
- Normal vector: n⃗ = ⟨a, b, c⟩
**Point-normal form:** a(x-x₀) + b(y-y₀) + c(z-z₀) = 0
**Distance from point (x₀,y₀,z₀) to plane:**
d = |ax₀ + by₀ + cz₀ - d| / √(a² + b² + c²)
**Parallel planes:** Same normal vector, different d.
**Angle between planes:** cos θ = |n₁⃗·n₂⃗| / (|n₁⃗||n₂⃗|)
Examples
Find normal to 2x + 3y - z = 5
Solution: n⃗ = ⟨2, 3, -1⟩