Discover Linear Algebra

Section 15.2 Terminology and notation

point-parallel form (of a line in $\R^n$)

the vector equation $\uvec{x} = \uvec{x}_0 + t \uvec{p}\text{,}$ where $\uvec{x}_0$ is a vector from the origin to a known point on the line, $\uvec{p}$ is a known parallel vector for the line, $\uvec{x}$ is a variable vector representing an arbitrary point on the line (again as a vector from the origin), and $t$ is a scalar parameter that varies as the arbitrary vector $\uvec{x}$ varies

point-parallel form (of a plane in $\R^n$)

the vector equation $\uvec{x} = \uvec{x}_0 + s \uvec{p}_1 + t \uvec{p}_2\text{,}$ where $\uvec{x}_0$ is a vector from the origin to a known point on the plane, $\uvec{p}_1,\uvec{p}_2$ are known parallel vectors for the plane that are not parallel to each other, $\uvec{x}$ is a variable vector representing an arbitrary point on the plane (again as a vector from the origin), and $s,t$ are scalar parameters that vary as the arbitrary vector $\uvec{x}$ varies

point-normal form (of a plane in $\R^3$)

the vector equation $\dotprod{\uvec{n}}{(\uvec{x} - \uvec{x}_0)} = 0\text{,}$ where $\uvec{x}_0$ is a vector from the origin to a known point on the plane, $\uvec{n}$ is a known normal vector for the plane, and $\uvec{x}$ is a variable vector representing an arbitrary point on the plane (again as a vector from the origin)