Position vector

The position vector of a point $P=(x,y,z)$ is the vector drawn from the origin to $P$:

$\mathbf{R}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}.$

Its magnitude is the distance from the origin: $|\mathbf{R}|=\sqrt{x^2+y^2+z^2}$.

Vectors in general are pure displacements: magnitude and direction but no anchor point. The position vector is the one exception. By convention its tail is always at the origin, so it specifies a location rather than a displacement. That pinning to the origin is what lets the same triple of numbers $(x,y,z)$ serve as both “the point $P$” and “the position vector of $P$.”

In other coordinate systems

Cylindrical $(r,\varphi ,z)$:

$\mathbf{R}=r\hat{\mathbf{r}}+z\hat{\mathbf{z}}.$

No $\varphi$ component, since the unit vector $\hat{\mathbf{r}}$ already encodes the azimuthal direction.

Spherical $(R,\theta ,\varphi )$:

$\mathbf{R}=R\hat{\mathbf{R}}.$

The whole position vector collapses to a single radial component. This is what makes spherical coordinates so clean for radially symmetric problems like the field of a point charge.

Why it matters in electromagnetics

Electrostatic and magnetostatic source equations are naturally written using position vectors. The electric field at observation point $\mathbf{R}$ from a charge $q$ located at source point ${\mathbf{R}}_1$ is

$\mathbf{E}=\frac{q(\mathbf{R}-{\mathbf{R}}_1)}{4\pi ϵ|\mathbf{R}-{\mathbf{R}}_1{|}^3}.$

The Distance vector $\mathbf{R}-{\mathbf{R}}_1$, the vector from source to observation point, is what carries the geometry. Position vectors are the bookkeeping that lets you write this cleanly even when neither the source nor the observation point sits at the origin.