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

The dot product of two vectors is a number. Many physical situations, however, produce a new direction out of two given ones: a wrench turned in a horizontal plane drives a screw vertically, and a charge moving horizontally through a vertical magnetic field is pushed sideways. The cross product describes such situations.

The cross product of $A$ and $B$ is the vector

A \times B = a_{x} A_{x} B_{x} a_{y} A_{y} B_{y} a_{z} A_{z} B_{z} = a_{x} (A_{y} B_{z} - A_{z} B_{y}) + a_{y} (A_{z} B_{x} - A_{x} B_{z}) + a_{z} (A_{x} B_{y} - A_{y} B_{x}), (2.1.8)

where the determinant is expanded along its first row. (In the expansion the middle term carries a minus sign, $- a_{y} (A_{x} B_{z} - A_{z} B_{x})$ , which we have written as $+ a_{y} (A_{z} B_{x} - A_{x} B_{z})$ .) The cross product is anticommutative,

A \times B = - B \times A, (2.1.9)

(interchanging two rows of a determinant changes its sign), it is distributive, $A \times (B + C) = A \times B + A \times C$ , but it is not associative: in general $A \times (B \times C) \neq = (A \times B) \times C$ . For the Cartesian unit vectors,

a_{x} \times a_{y} = a_{z}, a_{y} \times a_{z} = a_{x}, a_{z} \times a_{x} = a_{y}, a_{x} \times a_{x} = a_{y} \times a_{y} = a_{z} \times a_{z} = 0 . (2.1.10)

The cyclic order $x \to y \to z \to x$ is the definition of a right-handed coordinate system: a product of two unit vectors taken in this cyclic order gives $+$ the third one, taken against it gives $-$ the third one (Figure 2.1.6b). Multiplying out $(A_{x} a_{x} + A_{y} a_{y} + A_{z} a_{z}) \times (B_{x} a_{x} + B_{y} a_{y} + B_{z} a_{z})$ with (2.1.10) (and $a_{y} \times a_{x} = - a_{z}$ , etc.) reproduces the expanded form of (2.1.8).

Geometric meaning of the cross product

Using the components one can verify two facts (the details are in the "Going deeper" box below): Lagrange's identity $∣ A \times B ∣^{2} = A^{2} B^{2} - (A \cdot B)^{2} = A^{2} B^{2} (1 - cos^{2} θ_{A B}) = A^{2} B^{2} sin^{2} θ_{A B}$ , and $A \cdot (A \times B) = B \cdot (A \times B) = 0$ , i.e. $A \times B$ is perpendicular to both factors. Hence

Cross product: geometric form

A \times B = a_{n} A B sin θ_{A B}, (2.1.11)

where $a_{n}$ is the unit vector perpendicular to both $A$ and $B$ , in the direction in which a right-handed screw advances when $A$ is turned towards $B$ through the angle $θ_{A B}$ (right-hand rule).

The magnitude $A B sin θ_{A B}$ is the area of the parallelogram with sides $A$ and $B$ (Figure 2.1.5b): its base is $B$ and its height is $A sin θ_{A B}$ . So a cross product is an oriented area: a magnitude equal to an area and a direction normal to it. This is precisely how we shall represent surface elements, $d S = a_{n} d S$ . Parallel vectors ( $θ_{A B} = 0$ or $π$ ) have zero cross product, because the parallelogram collapses. In physics the cross product describes, for example, the torque $T = r \times F$ of a force about a point and the magnetic force $q u \times B$ on a moving charge.

Geometric meaning of the dot and cross products. (a) \vec A\cdot\vec B=B\,(A\cos\theta_{AB}): the length of \vec B times the projection of \vec A on it. — **Figure 2.1.5a.** Geometric meaning of the dot and cross products. (a) $A \cdot B = B (A cos θ_{A B})$ : the length of $B$ times the projection of $A$ on it.

$(b) \vec A\times\vec B is normal to the plane of \vec A and \vec B, and its length is the area of the parallelogram. The parallelogram lies in a horizontal plane seen obliquely from above. The curved arrow turns \vec A towards \vec B, and the right-hand rule then gives the (upward) direction of \vec A\times\vec B.$

Figure 2.1.5b. (b)

A \times B

is normal to the plane of

A

and

B

, and its length is the area of the parallelogram. The parallelogram lies in a horizontal plane seen obliquely from above. The curved arrow turns

A

towards

B

, and the right-hand rule then gives the (upward) direction of

A \times B

$The direction of a cross product. (a) \vec A and \vec B lie in the shaded horizontal plane. Curling the fingers of the right hand from \vec A towards \vec B (through the smaller angle), the thumb points along \vec A\times\vec B, and reversing the order reverses the result. (b) A memory aid for the unit vectors: going round the circle in the direction of the arrows gives a plus sign, going against it a minus sign.$

Figure 2.1.6. The direction of a cross product. (a)

A

and

B

lie in the shaded horizontal plane. Curling the fingers of the right hand from

A

towards

B

(through the smaller angle), the thumb points along

A \times B

, and reversing the order reverses the result. (b) A memory aid for the unit vectors: going round the circle in the direction of the arrows gives a plus sign, going against it a minus sign.

In plain words

$A \times B$ is a vector standing perpendicular to both $A$ and $B$ . Its length is the area of the parallelogram they span: largest when they are perpendicular, zero when they are parallel. Its direction comes from the right-hand rule, and swapping the factors flips it.

Common mistake

Getting the order or the middle sign wrong. $A \times B$ and $B \times A$ are opposite vectors, so the order of the factors must be kept exactly as written. When expanding the determinant, remember the minus sign in front of the $a_{y}$ term. A quick safeguard is to check at the end that the result is perpendicular to both factors, $A \cdot (A \times B) = 0$ and $B \cdot (A \times B) = 0$ .

Quick check

(i) Find $a_{z} \times a_{x}$ and $a_{y} \times a_{x}$ . (ii) For which angle between two vectors is $∣ A \times B ∣$ equal to $A \cdot B$ ?

(i) $a_{z} \times a_{x} = a_{y}$ (with the arrows of Figure 2.1.6b) and $a_{y} \times a_{x} = - a_{z}$ (against them). (ii) $A B sin θ_{A B} = A B cos θ_{A B}$ requires $tan θ_{A B} = 1$ , i.e. $θ_{A B} = 4 5^{\circ}$ .

Drag the two vectors below and watch their sum and difference, the dot product as a projection and the cross product as the area of the parallelogram: make them perpendicular to see $A \cdot B$ vanish, and parallel to see $A \times B$ vanish.

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2.1.5 The cross (vector) product

Geometric meaning of the cross product