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3D coordinates, lines & planes formulas
13 formulas for 3D coordinates, lines & planes, from the Coordinate Geometry section of the Quantix formula library. Tap 🧮 next to a formula to solve it for any variable with step-by-step working.
| Name | Formula | Note |
|---|---|---|
| Distance | √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) | |
| Direction cosines | l² + m² + n² = 1 🧮 Solve step by step | |
| Direction cosines from ratios | l = a/√(a²+b²+c²), … 🧮 Solve step by step | |
| Angle between lines | cos θ = |l₁l₂ + m₁m₂ + n₁n₂| | |
| Line — symmetric form | (x − x₁)/a = (y − y₁)/b = (z − z₁)/c 🧮 Solve step by step | |
| Line through two points | (x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−y₁) = (z−z₁)/(z₂−z₁) 🧮 Solve step by step | |
| Plane — general | ax + by + cz + d = 0 🧮 Solve step by step | normal (a, b, c) |
| Plane — intercept form | x/a + y/b + z/c = 1 🧮 Solve step by step | |
| Plane through a point | a(x − x₁) + b(y − y₁) + c(z − z₁) = 0 🧮 Solve step by step | |
| Point to plane distance | |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²) | |
| Angle between planes | cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (|n₁||n₂|) | |
| Sphere | (x − a)² + (y − b)² + (z − c)² = r² 🧮 Solve step by step | |
| Skew lines distance | |(b₁ × b₂)·(a₂ − a₁)| / |b₁ × b₂| |
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