Skip to content
Calculus II: Vector Calculus and Multivariable Analysis / Vectors and Curves in Space

Lines and Planes in ℝ³

The yield curve is a plane (or a collection of points on a surface) in multi-factor rate space; the constraint that a bond portfolio has zero DV01 defines a plane in weight space; a regression residual is the perpendicular distance from a data point to the fitted hyperplane. Behind all of these lies a single geometric principle: a plane in R3\mathbb{R}^3 is determined by a point and a normal vector, and a line is determined by a point and a direction vector. This note builds the Cartesian and parametric equations for both, derives the distance-from-a-point-to-a-plane formula via projection, and connects the machinery to the cross product from the previous lesson.
Learning objectives. By the end of this note you should be able to:
  • write the Cartesian equation Ax+By+Cz=DAx + By + Cz = D for a plane given a point and a normal vector, or given three non-collinear points;
  • write the parametric form r(t)=P0+tm\mathbf{r}(t) = \mathbf{P}_0 + t\mathbf{m} for a line given a point and a direction, and convert between parametric and symmetric forms;
  • find the angle between two planes from the angle between their normal vectors;
  • compute the distance from a point PP to a plane using the projection formula;
  • find the line of intersection of two planes by solving the system parametrically.

In this lesson

    1. Planes in ℝ³: The Normal-Vector Approach
    1. Parametric Form of a Plane
    1. Lines in ℝ³: Parametric and Symmetric Forms
    1. Angle between Two Planes
    1. Distance from a Point to a Plane
    1. Line of Intersection of Two Planes
    1. Summary Table

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.