Vector Equations of Lines and Planes

Taught in Math 115

Vector Equations of Lines §

Need 2 things to make a vector equation of the line

1. A point P on the line
2. A nonzero vector $\vec{d}$ in the direction of the line (called a direction vector for the line)

[!Definition 1.4.1]

A line in ${\mathbb{R}}^n$ through a point P with direction $\vec{d}$, where $\vec{d}\in {\mathbb{R}}^n$ is non-zero is given by vector equation $\vec{x}=\left[\begin{array}{c}{x}_1\\ ...\\ x_n\end{array}\right]=\overrightarrow{OP}+t\vec{d}$, t $\in \mathbb{R}$

The vector $\vec{d}$ is called a direction vector for this line

Vector Equation of a Line - Example §

Vector Equation of a Plane §

[!Vector Equation of a Plane]

A vector equation for a plane in ${\mathbb{R}}^n$ through a point P is given by

$\vec{x}=\left[\begin{array}{c}{x}_1\\ ...\\ x_n\end{array}\right]=\overrightarrow{OP}+s\vec{u}+t\vec{v},s,t,\in \mathbb{R}$

where $\vec{u},\vec{v}\in {\mathbb{R}}^n$ are non-zero non parallel vectors

Vector Equation of a Plane - Example §