Proposition. Show that the parametric equation $\overrightarrow {X}=\overrightarrow {X}+\overrightarrow {V}+t(\overrightarrow {U}-\overrightarrow {V})$ represents the line through $\overrightarrow {U}$ and $\overrightarrow {V}$ if $\overrightarrow {U}$ and $\overrightarrow {V}$ are any two vectors which are not equal.
Can you give a hint to prove?
Hint:
The correct equation is $\vec X=\vec V +t(\vec U-\vec V)$, with $t \in \mathbb{R}$. Note that for $t=0$ we have $\vec X= \vec V$ and for $t=1$ we have $\vec X=\vec U$.
And for the other values of $t$ we have a vector $\vec X$ that points to a point of the line that passes thorough the two points pointed by the vectors $\vec U$ and $\vec V$.
(Note that here a vector is pointing to a point).