Distance vector

The distance vector from point $P_1$ to point $P_2$ is the vector you’d add to ${\mathbf{R}}_1$ to land at ${\mathbf{R}}_2$:

${\mathbf{R}}_{12}={\mathbf{R}}_2-{\mathbf{R}}_1=(x_2-x_1)\hat{\mathbf{x}}+(y_2-y_1)\hat{\mathbf{y}}+(z_2-z_1)\hat{\mathbf{z}},$

where ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ are the position vectors of $P_1$ and $P_2$. Its magnitude is the distance between the two points:

$|{\mathbf{R}}_{12}|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$

The direction is “from $P_1$ to $P_2$.” Reversing the order flips the sign: ${\mathbf{R}}_{21}=-{\mathbf{R}}_{12}$.

In cylindrical coordinates

For $P_1=(r_1,{\varphi }_1,z_1)$ and $P_2=(r_2,{\varphi }_2,z_2)$, the distance between them is

$|{\mathbf{R}}_{12}|=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos ({\varphi }_2-{\varphi }_1)+(z_2-z_1{)}^2}.$

This is the 3D law of cosines: $r_1^2+r_2^2-2r_1{r}_2\cos \Delta \varphi$ gives the squared horizontal separation, and $(z_2-z_1{)}^2$ adds the vertical separation in quadrature.

In spherical coordinates

For $P_1=(R_1,{\theta }_1,{\varphi }_1)$ and $P_2=(R_2,{\theta }_2,{\varphi }_2)$:

$|{\mathbf{R}}_{12}{|}^2=R_1^2+R_2^2-2R_1{R}_2[\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos ({\varphi }_2-{\varphi }_1)].$

The bracket is $\cos \gamma$, where $\gamma$ is the angle subtended at the origin between the two position vectors — the spherical law of cosines.

Why it matters in electromagnetics

The distance vector from a source charge to an observation point is the natural variable in Coulomb’s law, the Biot-Savart law, and the integral expressions for field due to distributed sources:

$\mathbf{E}(\mathbf{R})=\frac{1}{4\pi ϵ}{\int }_{V^{\prime }}{\rho }_v({\mathbf{R}}^{\prime })\frac{\mathbf{R}-{\mathbf{R}}^{\prime }}{|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^3}d{v}^{\prime }.$

The integrand walks the source point ${\mathbf{R}}^{\prime }$ through the source region while $\mathbf{R}$ stays fixed; the distance vector $\mathbf{R}-{\mathbf{R}}^{\prime }$ aims from each source bit to the observation point.