I'm sorry if this is a dumb question but I've never heard a convincing explanation for why seemingly all of physics names the position vector "$\vec{r}$". I've tried translating it into just about every language that would make sense, and I've asked several professors at my college, but I can't find a satisfactory answer. Was it just something somebody decided one day?
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4 Answers
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The $r$ is for "radius", and in particular, describes the radial vector from the origin to the location described by the vector. This is sensible because some sort of polar or spherical coordinates are the most common for many physical applications, where the forces described have some sort of spherical symmetry, and point radially outward.
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As far as I know, the letter $r$ comes from the word "radius." This is best seen in polar coordinates, where the position vector of an object is given by $r\,\hat{r}$, where $r$ is the distance from the origin to the object (ie the radius of the circle that passes through our object and is concentric with the origin) and $\hat{r}$ is the radial basis vector in polar coordinates. Since we write $r\,\hat{r}$, we might as well use another $r$ to write the expression for the position of an object in polar coordinates as
$$\vec{r}=r\,\hat{r}.$$
As far as I know, this is the origin of this notation. Often people will write $\vec{x}$, but they don't want to confuse it with the $x$-direction in a particular problem (or they've already used that letter), and so they use the next thing they know is often used to describe a position, and thus we have $\vec{r}$.
I hope this helps!
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It can be motivated by considering central force motions, like around the sun. The force points against the radial vector $\vec r$ from the sun to the heavenly body in question.
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$\vec{r}$ is actually a radius vector but if we consider center of our circle as origin of our x-y axes then the radius vector will also denote the position of particle performing circular motion that's why we can say that $\vec{r}$ is position vector but if we consider origin as any other point in space then $\vec{r}$ won't denote position vector.
