How do you find the parametric equation of a line in $\displaystyle R^4$ that passes through the points (1,2,3,4) and (5,6,7,8)?
Well, the general form of a line in any dimensions looks like this:
$\displaystyle \mathbf{y}=t\,\mathbf{v}+\mathbf{b},$
where $\displaystyle \mathbf{v}$ is a vector parallel to the line, and $\displaystyle \mathbf{b}$ is a vector from the origin to a point on the line. How could you get $\displaystyle \mathbf{v}$ and $\displaystyle \mathbf{b}?$