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 — they have 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. This 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 $(r,\varphi ,z)$:

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

Note there is no $\varphi$ component — 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.