Question

What is the form of vector and parametric equations of lines in $R^{2}$ and $R^{3}$

Answer 1

The variable $t$ (called a parameter) varies from $-\infty$ to $\infty$, the point $\mathbf{x}$ traces out the line $L$.

Accordingly, we have the following result. Let $L$ be the line in $R^{2}$ or $R^{3}$ that contains the point $\mathbf{x}_{0}$ and is parallel to the nonzero vector $\mathbf{v}$.

Then the equation of the line through $\mathbf{x}_{0}$ that is parallel to $\mathbf{v}$ is $$ \mathbf{x}=\mathbf{x}_{0}+t \mathbf{v} $$ If $\mathbf{x}_{0}=\mathbf{0}$, then the line passes through the origin and the equation has the form $$ \mathbf{x}=t \mathbf{v} $$