Differences between Vector magnitude and distance vector

Question

I will be so grateful if someone can explain me in a simple language what is difference between vector magnitude namely $$r=\left|\vec{r}_1-\vec{r}_2\right|$$ and distance vector namely $$\vec r_{12}=\vec r_1-\vec r_2$$ or introduce a good reference

Answer 1

Let's take an example in the plane. Let $\vec{a} = (1,0)$ and $\vec{b} = (0,1)$. Then the difference vector is $$\vec{r}_{ab} = \vec{b} - \vec{a} = (0,1) - (1,0) = (-1,1).$$ In other words, if you are walking from $a$ and want to go to $b$, you need to make one step to the left and another step up.

Finally, note that $$ r = \left|\vec{r}_{ab}\right| = \sqrt{(-1)^2 + 1^2} = \sqrt2, $$ so if you are walking from $a$ to $b$ directly, on a straight line segment, you will have to walk a grand total of $\sqrt2$ units.

Answer 2

Your $r$ is the length of the distance from two points indicated by the vectors $r_1$ and $r_2$.

$\vec r_{12}$ is a vector, this means that it is not only the length of the distance from two points but also the orientation of this distance, directed from $r_1$ to $r_2$