$\hat{\mathbf{u}} = \frac{\mathbf{u}}{|\mathbf{u}|}$

$\hat{\mathbf{r}} = 3\hat{\mathbf{i}} + 2\hat{\mathbf{j}}$
- Example 1: A position vector (or $\mathbf{r} = 3\mathbf{i} + 2\mathbf{j}$) is one whose x-component is 3 units and y-component is 2 units (SI units: meters).
Example 2: A velocity vector $\mathbf{v}=3t\hat{\mathbf{i}}-4\hat{\mathbf{j}}$
- The velocity has an x-component of $3t$ units (it varies with time) and a y-component of -4 units (it is constant). (SI units: m/s)
Problem: Specify the unit vector extending from the origin towards the point G (2, -2, -1)?
- Solution: $\mathbf{G} = 2\mathbf{a}_x - 2\mathbf{a}_y - \mathbf{a}_z$
- $\hat{\mathbf{r}} = \frac{x\hat{\mathbf{x}} + y\hat{\mathbf{y}} + z\hat{\mathbf{z}}}{\sqrt{x^2+y^2+z^2}}$
- $|\mathbf{G}| = \sqrt{(2)^2 + (-2)^2 + (-1)^2} = 3$
- $\mathbf{a}_G = \frac{\mathbf{G}}{|\mathbf{G}|} = \frac{2}{3}\mathbf{a}_x - \frac{2}{3}\mathbf{a}_y - \frac{1}{3}\mathbf{a}_z = 0.667\mathbf{a}_x - 0.667\mathbf{a}_y - 0.333\mathbf{a}_z$

The location of a point in three-dimensions (3-D) can be described by listing its Cartesian coordinates (x, y, z).
The vector to that point from the origin (O) is called the position vector.
The distance from the origin $|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$
Unit vector pointing radially outward: $\hat{\mathbf{r}} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{x\hat{\mathbf{x}} + y\hat{\mathbf{y}} + z\hat{\mathbf{z}}}{\sqrt{x^2+y^2+z^2}}$
The infinitesimal (very small) displacement vector from (x, y, z) to (x + dx, y + dy, z + dz), is $\mathbf{dl} = dx\hat{\mathbf{x}} + dy\hat{\mathbf{y}} + dz\hat{\mathbf{z}}$

In electrodynamics, one frequently encounters problems involving two points.
A source point, $\mathbf{r}'$, where an electric charge is located, and a field point, $\mathbf{r}$, at which we are calculating the electric or magnetic field.
It pays to adopt right from the start some short-hand notation for the separation vector from the source point to the field point, $\boldsymbol{\varkappa} = \mathbf{r} - \mathbf{r}'$
Its magnitude is $|\boldsymbol{\varkappa}| = |\mathbf{r}-\mathbf{r}'|$
Unit vector in the direction from $\mathbf{r}'$ to $\mathbf{r}$ is $\hat{\boldsymbol{\varkappa}} = \frac{\boldsymbol{\varkappa}}{|\boldsymbol{\varkappa}|} = \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|}$

$\boldsymbol{\varkappa} = (x - x')\hat{\mathbf{x}} + (y - y')\hat{\mathbf{y}} + (z - z')\hat{\mathbf{z}}$
$|\boldsymbol{\varkappa}|= \sqrt{(x - x')^2 + (y - y')^2 + (z - z')^2}$
$\hat{\boldsymbol{\varkappa}} = \frac{(x - x')\hat{\mathbf{x}} + (y - y')\hat{\mathbf{y}} + (z - z')\hat{\mathbf{z}}}{\sqrt{(x - x')^2 + (y - y')^2 + (z - z')^2}}$

The rectangular coordinate system consists of two real number lines that intersect at a right angle.
The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
These two number lines define a flat surface called a plane, and each point