Coordinate System
- Cartesian Coordinate System: Points are represented by (x, y, z) where x, y, and z are distances along the x, y, and z axes.
- Distance between two points: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Lines in 3D
- Vector equation: r = a + λb, where r is the position vector of any point on the line, a is the position vector of a point on the line, b is a vector parallel to the line, and λ is a scalar parameter.
- Cartesian equation: (x - x₁) / a = (y - y₁) / b = (z - z₁) / c, where (x₁, y₁, z₁) is a point on the line and (a, b, c) are the direction ratios of the line.
Planes in 3D
- Vector equation: r·n = d, where r is the position vector of any point on the plane, n is a vector normal to the plane, and d is the perpendicular distance of the plane from the origin.
- Cartesian equation: ax + by + cz = d, where (a, b, c) are the direction ratios of the normal to the plane and d is the perpendicular distance of the plane from the origin.
Angle between Lines and Planes
- Angle between two lines: cos θ = (a₁a₂ + b₁b₂ + c₁c₂) / √(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²)
- Angle between a line and a plane: sin θ = |a₁l + b₁m + c₁n| / √(a₁² + b₁² + c₁²)√(l² + m² + n²)
Distance between Points, Lines, and Planes
- Distance between two points: See the coordinate system section.
- Distance between a point and a line: d = |(b × (a - c))| / |b|
- Distance between a point and a plane: d = |ax₁ + by₁ + cz₁ - d| / √(a² + b² + c²)
Applications
- Vector algebra
- Calculus
- Physics (e.g., kinematics, dynamics)
- Engineering (e.g., structural analysis, computer graphics)
Note: These are just brief notes. For more detailed explanations and examples, refer to textbooks or online resources.