Constructing a vector from a point

[Mr Davis 97]

So I have solved the following problem: Consider two points located at $\vec{r}_1$ and $\vec{r}_2$, separated by distance $r = |\vec{r}_1 - \vec{r}_2|$. Find a vector $\vec{A}$ from the origin to a point on the line between $\vec{r}_1$ and at $\vec{r}_2$ at a distance $xr$ from the point at $\vec{r}_1$ where $x$ is some number.

So I have solved this problem. The answer is $\vec{A} = (1 - x) \vec{r}_1 + x \vec{r}_2$. I did this by allowing $\vec{r}$ to vary with $x$, and then $\vec{A}$ was just the vector sum of $\vec{r}_1$ and $x \vec{r}$. However, I am not really understanding the solution. When we substitute 0 for x, we find that we get $\vec{A} = \vec{r}_1$, as expected. However, when we substitute r for x, we don't get $\vec{r}_2$. What is going on here?

 

[pwsnafu]

$A = r_2$ when $x = 1$. The question says "at a distance $xr$ from $r_1$", so when $x=1$ you are distance r from $r_1$ which is $r_2$.

 

[Mr Davis 97]

$A = r_2$ when $x = 1$. The question says "at a distance $xr$ from $r_1$", so when $x=1$ you are distance r from $r_1$ which is $r_2$.

Oh wow, that is pretty obvious now. Thanks!