Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

In a Cartesian (rectangular) coordinate system the three mutually perpendicular unit vectors $a_{x}$ , $a_{y}$ , $a_{z}$ point in the directions of increasing $x$ , $y$ and $z$ . A unit vector is a vector of length one, so it carries only a direction. Any vector can be written as the sum of three vectors along these directions,

A B = A_{x} a_{x} + A_{y} a_{y} + A_{z} a_{z}, = B_{x} a_{x} + B_{y} a_{y} + B_{z} a_{z}, (2.1.1)

where the scalars $A_{x}$ , $A_{y}$ , $A_{z}$ are the components of $A$ (Figure 2.1.2). You can think of the components as walking instructions: "go $A_{x}$ along $x$ , then $A_{y}$ along $y$ , then $A_{z}$ along $z$ ". Placing the three vectors $A_{x} a_{x}$ , $A_{y} a_{y}$ and $A_{z} a_{z}$ head to tail brings us from the tail of $A$ to its head, and they form the edges of a rectangular box whose main diagonal is $A$ .

A vector \vec A and its Cartesian components. The three vectors A_x\vec a_x, A_y\vec a_y, A_z\vec a_z placed head to tail add up to \vec A, which is the main diagonal of the dashed box. The unit vectors \vec a_x,\vec a_y,\vec a_z (drawn in black at the origin O) have length one and the same directions everywhere in space. — **Figure 2.1.2.** A vector $A$ and its Cartesian components. The three vectors $A_{x} a_{x}$ , $A_{y} a_{y}$ , $A_{z} a_{z}$ placed head to tail add up to $A$ , which is the main diagonal of the dashed box. The unit vectors $a_{x}, a_{y}, a_{z}$ (drawn in black at the origin O) have length one and the same directions everywhere in space.

Magnitude

Applying Pythagoras' theorem twice (first to the right triangle in the $x y$ -plane with legs $∣ A_{x} ∣$ and $∣ A_{y} ∣$ , whose hypotenuse is $A_{x}^{2} + A_{y}^{2}$ , then to the vertical right triangle whose legs are that hypotenuse and $∣ A_{z} ∣$ ) gives the length of the box diagonal:

A = A = A_{x}^{2} + A_{y}^{2} + A_{z}^{2} . (2.1.2)

The magnitude is never negative, and it is zero only for the zero vector $0$ .

Unit vector

Dividing a vector by its own magnitude leaves its direction unchanged and makes its length equal to one. The unit vector in the direction of $A$ is therefore

a_{A} A = \frac{A}{A} = \frac{A _{x} a _{x} + A _{y} a _{y} + A _{z} a _{z}}{A _{x}^{2} + A _{y}^{2} + A _{z}^{2}}, = A a_{A} . (2.1.3)

The second form, "magnitude times unit vector", separates the two pieces of information carried by a vector, and it is the form in which most field expressions of this course are written: for example the field of a point charge will be (a number) $\times a_{R}$ (Section 3.1). A unit vector is dimensionless, and the unit of $A$ is carried by its magnitude.

In plain words

A vector is "how much" times "which way". The magnitude $A$ answers "how much" (with the unit), the unit vector $a_{A}$ answers "which way" (with no unit).

2.1.2 Components, magnitude and unit vector

Magnitude

Unit vector