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# Lessons 1-4: Vectors, Lines, and Planes in Space & Quadratic Surfaces ## Review of Vectors ### Vector Operations * **Addition/Subtraction:** `` + `` = `` * **Scalar Multiplication:** n`` = `` * **Magnitude of a vector:** ||v|| = √(v₁² + v₂² + v₃²) ### Position Vectors * **Vector from point A...

# Lessons 1-4: Vectors, Lines, and Planes in Space & Quadratic Surfaces ## Review of Vectors ### Vector Operations * **Addition/Subtraction:** `` + `` = `` * **Scalar Multiplication:** n`` = `` * **Magnitude of a vector:** ||v|| = √(v₁² + v₂² + v₃²) ### Position Vectors * **Vector from point A (x₁, y₁, z₁) to point B (x₂, y₂, z₂):** `` ### Equations of geometric Shapes * **Sphere:** (x-h)² + (y-k)² + (z-l)² = r² (where (h, k, l) is the center and r is the radius) ### Dot Product of Two Vectors * **Dot Product:** ` ⋅ ` = v₁u₁ + v₂u₂ + v₃u₃ (resulting in a real number) ### Angle between Vectors * **Cosine of the angle:** cos θ = (v ⋅ u) / (||v|| ||u||) ### Orthogonal Projections * **Projection of a onto b:** projba = (a ⋅ b) / ||b||2 * b ### Cross Product * **Cross Product:** u × v = | i j k | | u₁ u₂ u₃ | | v₁ v₂ v₃ | * **Area of a parallelogram:** ||u × v|| ### Two Vectors are Orthogonal if... * Their dot product equals 0 => u ⋅ v = 0 ## Lines and Planes in Space ### Equations of Lines and Line Segments * **Parametric Equations:** x = x₁ + at, y = y₁ + bt, z = z₁ + ct (where (x₁, y₁, z₁) is a point on the line and is the direction vector) * **Symmetric Equations:** x - x₁ / a = y - y₁ / b = z - z₁ / c ### Equations of Planes * **Plane equation:** Ax + By + Cz = D * **Parallel planes:** If normal vectors are parallel * **Orthogonal planes:** If normal vectors are orthogonal * **Intersecting planes:** If normal vectors are linearly independent ## Quadratic Surfaces | Surface | Equation | |-------------------|------------------------------------------------| | Cylinder | x² / a² + y² / b² = 1 | | Ellipsoid | x² / a² + y² / b² + z² / c² = 1 | | Elliptic Paraboloid | x² / a² + y² / b² = z | | Hyperbolic Paraboloid | x² / a² - y² / b² = z | | Hyperboloid of one sheet | x² / a² + y² / b² - z² / c² = 1 | | Hyperboloid of two sheets | x² / a² - y² / b² - z² / c² = 1 |