Equations of Lines and Planes in R² and R³

This cheat sheet covers the various ways to represent lines and planes in two and three-dimensional space, including vector, parametric, and scalar equations, and explores their intersections and relationships.

Core Principles

Lines in R² can be defined by slope-intercept, standard, vector, or parametric equations.
Lines in R³ can be defined by vector, parametric, or symmetric equations, but not scalar equations.
Planes in R³ are defined by vector, parametric, or scalar equations.
The intersection of lines and planes can result in one solution, infinite solutions, or no solution.
Systems of linear equations can be solved using elimination/substitution or matrix methods (echelon form).

Action Steps

To find the vector equation of a line, identify a point on the line (position vector $ \vec{r_0} $) and a direction vector parallel to the line ($\vec{m}$).
To find the parametric equations of a line, use the components of the position vector and direction vector.
To find the scalar equation of a plane, use the normal vector and a point on the plane, or two direction vectors and a point on the plane.
To solve a system of linear equations, use elimination/substitution or matrix methods (Gaussian elimination to reach echelon form).
Check for parallel lines by comparing their direction vectors; check for parallel planes by comparing their normal vectors.

Formulas

Vector Equation of a Line (R²): $ \vec{r} = \vec{r_0} + t\vec{m} $ or $ [x, y] = [x_0, y_0] + t[m_1, m_2] $
Parametric Equation of a Line (R²): $ x = x_0 + tm_1, y = y_0 + tm_2 $
Vector Equation of a Line (R³): $ \vec{r} = \vec{r_0} + t\vec{m} $ or $ [x, y, z] = [x_0, y_0, z_0] + t[m_1, m_2, m_3] $
Symmetric Equation of a Line (R³): $ \frac{x-x_0}{m_1} = \frac{y-y_0}{m_2} = \frac{z-z_0}{m_3} $
Scalar Equation of a Plane (R³): $ Ax + By + Cz + D = 0 $
Vector Equation of a Plane (R³): $ \vec{r} = \vec{r_0} + t\vec{a} + s\vec{b} $ or $ [x, y, z] = [x_0, y_0, z_0] + t[a_1, a_2, a_3] + s[b_1, b_2, b_3] $
Parametric Equation of a Plane (R³): $ x = x_0 + ta_1 + sb_1, y = y_0 + ta_2 + sb_2, z = z_0 + ta_3 + sb_3 $
Distance between skew lines: $ d = \frac{|P_1P_2 \cdot \vec{n}|}{|\vec{n}|} $ where $ \vec{n} = \vec{m_1} \times \vec{m_2} $
Distance from a point to a plane: $ d = \frac{|PQ \cdot \vec{n}|}{|\vec{n}|} $

Key Terms

Position Vector: A vector that starts at the origin and ends at a specific point.
Direction Vector: A vector parallel to a line or plane, indicating its direction.
Normal Vector: A vector perpendicular to a plane or line.
Scalar Equation: An equation representing a line (R²) or plane (R³) using coefficients and constants.
Vector Equation: An equation representing a line or plane using vector addition and scalar multiplication.
Parametric Equation: A set of equations that express the coordinates of points on a line or plane in terms of one or more parameters.
Echelon Form: A form of a matrix where each successive row has its first non-zero element further to the right than the previous row.
Consistent System: A system of equations with at least one solution.
Inconsistent System: A system of equations with no solution.
Skew Lines: Lines in 3D space that are neither parallel nor intersecting.

Timeline

Unit 6: Equations of Lines in R²
Unit 8, Lesson 2: Equations of Lines in R³
Unit 8, Lesson 3: Equations of Planes in Vector and Parametric Form
Unit 8, Lesson 4: Equations of Planes in Scalar Form
Unit 9, Lesson 5: Intersection of Lines in R² and R³
Unit 8, Lesson 6: Intersection of Lines and Planes in R³
Unit 9, Lesson 7: System of Linear Equations (Matrices)
Unit 9, Lesson 8: Intersection of Planes in R³