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

We often need to know "how much of one vector points along another". How much of the force on a sled actually pulls it forwards? How much of the electric field crosses a surface instead of sliding along it? The dot product answers exactly this kind of question.

The dot product of two vectors is the scalar

A \cdot B A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}, = B \cdot A . (2.1.5)

It is commutative and distributive, $A \cdot (B + C) = A \cdot B + A \cdot C$ , and the dot product of a vector with itself is the square of its magnitude, $A \cdot A = A_{x}^{2} + A_{y}^{2} + A_{z}^{2} = A^{2}$ . For the Cartesian unit vectors,

a_{x} \cdot a_{x} = a_{y} \cdot a_{y} = a_{z} \cdot a_{z} = 1, a_{x} \cdot a_{y} = a_{y} \cdot a_{z} = a_{z} \cdot a_{x} = 0, (2.1.6)

and indeed (2.1.5) follows from these relations by multiplying out $(A_{x} a_{x} + A_{y} a_{y} + A_{z} a_{z}) \cdot (B_{x} a_{x} + B_{y} a_{y} + B_{z} a_{z})$ term by term: of the nine products only the three "diagonal" ones survive, because the six "mixed" products such as $a_{x} \cdot a_{y}$ are zero.

Geometric meaning of the dot product

Let $θ_{A B}$ ( $0 \leq θ_{A B} \leq π$ ) be the angle between $A$ and $B$ when they are drawn from a common point. We compute the length of the third side of the triangle they form, $A - B$ (Figure 2.1.3c), in two ways.

With the law of cosines of plane geometry: $∣ A - B ∣^{2} = A^{2} + B^{2} - 2 A B cos θ_{A B}$ .
With the dot product and its distributive law: $∣ A - B ∣^{2} = (A - B) \cdot (A - B) = A \cdot A - 2 A \cdot B + B \cdot B = A^{2} + B^{2} - 2 A \cdot B$ .

The two right-hand sides must be equal, and comparing them gives

Dot product: geometric form

A \cdot B = A B cos θ_{A B} . (2.1.7)

Thus the dot product is the product of the length of one vector and the projection of the other onto it (Figure 2.1.5a): $A \cdot a_{B} = A cos θ_{A B}$ is the scalar component of $A$ in the direction of $B$ . In particular each Cartesian component is a projection, $A_{x} = A \cdot a_{x}$ , $A_{y} = A \cdot a_{y}$ , $A_{z} = A \cdot a_{z}$ . The sign of the dot product tells us at a glance whether two vectors point "roughly the same way", are perpendicular, or point "roughly opposite ways" (Figure 2.1.4). Three consequences are used constantly:

$A \cdot B = 0$ (with $A, B \neq = 0$ ) means $A ⊥ B$ ,
the angle between two vectors is $θ_{A B} = arccos (A \cdot B / A B)$ , and
only the part of a vector along a direction contributes to a dot product. The work done by a force over a small displacement $d ℓ$ is $d W = F \cdot d ℓ$ , and the flux of a field through a small surface element is $A \cdot d S$ . Both of these appear in this chapter.

The sign of the dot product. The green bar is the projection A\cos\theta_{AB} of \vec A on the direction of \vec B. (a) Acute angle: the projection points along \vec B and \vec A\cdot\vec B>0. (b) Right angle: the projection shrinks to a point and \vec A\cdot\vec B=0. (c) Obtuse angle: the projection points backwards and \vec A\cdot\vec B<0. — **Figure 2.1.4.** The sign of the dot product. The green bar is the projection $A cos θ_{A B}$ of $A$ on the direction of $B$ . (a) Acute angle: the projection points along $B$ and $A \cdot B > 0$ . (b) Right angle: the projection shrinks to a point and $A \cdot B = 0$ . (c) Obtuse angle: the projection points backwards and $A \cdot B < 0$ .

In plain words

$A \cdot B$ measures how much $A$ and $B$ "agree in direction", weighted by their lengths. It is largest ( $A B$ ) when they are parallel, zero when they are perpendicular and most negative ( $- A B$ ) when they are antiparallel. A rope pulling a sled at an angle $θ$ to the ground does work $F d ℓ cos θ$ per step $d ℓ$ : only the forward part of the pull counts.

Quick check

Without computing any angle, decide whether $A = a_{x} + a_{y}$ and $B = a_{x} - a_{y}$ are perpendicular. What is the scalar component of $A$ along $a_{z}$ ?

$A \cdot B = (1) (1) + (1) (- 1) + (0) (0) = 0$ , so they are perpendicular (they are the two diagonals of a square). The scalar component along $a_{z}$ is $A \cdot a_{z} = A_{z} = 0$ : $A$ lies in the $x y$ -plane.

2.1.4 The dot (scalar) product

Geometric meaning of the dot product