MODULUS / REFERENCE
Formulas: Analytic geometry
Complete Android Modulus reference: formulas for Analytic geometry.
Matrices and determinants
• det A = ad − bc for a 2 × 2 matrix
• A−1 exists when det A ≠ 0
• For Ax = b, the solution is x = A−1b when det A ≠ 0
Vectors
• |a| = √(x² + y² + z²)
• a · b = x1x2 + y1y2 + z1z2
• cos φ = (a · b) / (|a||b|)
• a × b = (y1z2 − z1y2, …)
Lines and planes
• Line: Ax + By + C = 0
• Plane: Ax + By + Cz + D = 0
• d = √((x2 − x1)² + (y2 − y1)²)
• d(point, line) = |Ax0 + By0 + C| / √(A² + B²)
Surfaces and conics
• Circle: (x − a)² + (y − b)² = R²
• Ellipse: x² / a² + y² / b² = 1
• Parabola: y² = 2px or x² = 2py
• Hyperbola: x² / a² − y² / b² = 1
• det A = ad − bc for a 2 × 2 matrix
• A−1 exists when det A ≠ 0
• For Ax = b, the solution is x = A−1b when det A ≠ 0
Vectors
• |a| = √(x² + y² + z²)
• a · b = x1x2 + y1y2 + z1z2
• cos φ = (a · b) / (|a||b|)
• a × b = (y1z2 − z1y2, …)
Lines and planes
• Line: Ax + By + C = 0
• Plane: Ax + By + Cz + D = 0
• d = √((x2 − x1)² + (y2 − y1)²)
• d(point, line) = |Ax0 + By0 + C| / √(A² + B²)
Surfaces and conics
• Circle: (x − a)² + (y − b)² = R²
• Ellipse: x² / a² + y² / b² = 1
• Parabola: y² = 2px or x² = 2py
• Hyperbola: x² / a² − y² / b² = 1
Values are provided for reference. Check the required unit system and conditions before engineering or laboratory use.