Overview
This lecture covers equations of lines and planes in three-dimensional space, including vector, parametric, and symmetric forms, as well as intersections, distances, and relationships between lines and planes.
Equations of Lines in 3D
- A line in 3D is determined by a point and a direction vector.
- Vector equation of a line: r = r₀ + t·v, where r₀ is a position vector to a point on the line, v is a direction vector, and t is a real parameter.
- Parametric equations: x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where (a, b, c) are the components of v.
- Symmetric equations: (x - x₀)/a = (y - y₀)/b = (z - z₀)/c, if none of a, b, c are zero.
- Direction numbers (a, b, c) indicate the direction of the line and are any proportional set to the direction vector.
Examples & Line Segments
- To find a line through two points, use the vector difference for the direction.
- A line segment between two points A and B is represented by r(t) = (1 - t)·r₀ + t·r₁, with 0 ≤ t ≤ 1.
Equations of Planes
- A plane in 3D is defined by a point and a normal vector.
- Vector equation: n·(r - r₀) = 0, where n is the normal vector, r₀ is a vector to a point on the plane.
- Scalar equation: a(x - x₀) + b(y - y₀) + c(z - z₀) = 0; simplifies to linear form ax + by + cz + d = 0.
- The normal vector (a, b, c) is perpendicular to the plane.
- The distance from a point (x₁, y₁, z₁) to a plane is |a·x₁ + b·y₁ + c·z₁ + d| / sqrt(a² + b² + c²).
Intersections and Angles
- Two planes are parallel if their normal vectors are proportional.
- The intersection of two non-parallel planes is a line; the angle between planes equals the angle between their normals, computed using the dot product.
- The intersection line can be found using the cross product of the normals and a point common to both planes.
Distances and Skew Lines
- Skew lines are lines that do not intersect and are not parallel.
- The distance between skew lines is the distance between parallel planes containing each line.
Key Terms & Definitions
- Vector equation of a line — Equation expressing all points on a line as r = r₀ + t·v.
- Parametric equations — Component-wise equations for a line involving a parameter t.
- Symmetric equations — Form equating differences in each coordinate divided by direction numbers.
- Direction numbers — The components of a vector indicating a line’s direction.
- Normal vector — A vector perpendicular to a plane.
- Scalar equation of a plane — An equation of the form ax + by + cz + d = 0.
- Skew lines — Lines in space that do not intersect and are not parallel.
Action Items / Next Steps
- Review textbook sections on vector equations, parametric equations, and planes (James Stewart, Early Transcendentals, Ch. 12).
- Practice finding equations of lines/planes through given points and with given direction/normal vectors.
- Complete exercises on intersections, distances, and angles between lines and planes.