Vectors, Fields, and Coordinate Geometry

ED02: Vectors, Fields, and Coordinate Geometry

Electrodynamics is written in the language of vector fields. Before deriving Coulomb’s law, Gauss’s Law, the Biot-Savart law, or Maxwell’s equations in detail, we need a precise geometric language for direction, magnitude, components, position, and spatial variation.

The central lesson of this article is that a vector is not the same thing as its list of components. The vector is the geometric object. Components are the numbers used to describe that object relative to a chosen basis.

That distinction becomes especially important in cylindrical and spherical coordinates, where the basis directions themselves change from point to point.

This article therefore develops the coordinate geometry that later electrodynamics will repeatedly use:

• vectors and scalars,
• basis vectors and components,
• dot and cross products,
• position and displacement vectors,
• scalar and vector fields,
• orthogonal curvilinear coordinates,
• scale factors,
• Cartesian, cylindrical, and spherical bases,
• and differential line, area, and volume elements.

Gradient, divergence, and curl are intentionally deferred to ED03. Here the emphasis is the geometry from which those operators will later be derived.

1 Scalars and vectors

A scalar is a quantity specified by a magnitude, possibly with units, but without an intrinsic spatial direction.

Examples include

(1)

A vector has both magnitude and direction.

Examples include

(2)

A geometric vector can be represented by a directed line segment. If A is a vector, its magnitude is written

(3)

The zero vector has zero magnitude:

(4)

Unlike an ordinary number, a vector cannot in general be characterized by magnitude alone.

2 Vectors versus components

Choose an orthonormal Cartesian basis

(5)

Any vector A can be written

(6)

The numbers

(7)

are the components of A in that basis.

If we rotate the coordinate axes, the vector itself does not change, but the components generally do.

This is a fundamental distinction:

(8)

The same physical Electric Field can therefore have different component triples in different coordinate systems while representing the same field at the same physical point.

Magnitude from components

Because the Cartesian basis is orthonormal,

(9)

with analogous relations for the other basis vectors.

Therefore

(10)

so

(11)

3 Vector addition and scalar multiplication

If

(12)

and

(13)

then

(14)

For scalar c,

(15)

Vector addition is geometric. Components make the calculation convenient, but the parallelogram or head-to-tail rule gives the underlying geometric meaning.

4 The dot product

For vectors A and B separated by angle 𝜃,

(16)

The dot product produces a scalar.

In an orthonormal Cartesian basis,

(17)

Projection

The scalar component of A along B is

(18)

where

(19)

The corresponding vector projection is

(20)

Equivalently,

(21)

This same projection idea will later appear in electric flux, magnetic flux, work, boundary conditions, polarization, and wave decomposition.

Orthogonality

If two nonzero vectors are perpendicular,

(22)

then

(23)

Thus the dot product is an algebraic test for orthogonality.

5 The cross product

The cross product is defined geometrically by

(24)

with direction perpendicular to the plane containing A and B according to the right-hand rule.

Thus

(25)

where n is the oriented unit Normal.

The order matters:

(26)

For a right-handed Cartesian basis,

x ×y	= z,	(27)
y ×z	= x,	(28)
z ×x	= y.	(29)

Reversing any pair changes the sign.

Cartesian component form

Expanding bilinearly gives

A × B =	(AyBz − AzBy)x	(30)
	+ (AzBx − AxBz)y	(31)
	+ (AxBy − AyBx)z.	(32)

This expression should not obscure the geometric meaning: the cross product represents oriented area and a normal direction.

6 Triple products

The scalar triple product is

(33)

Its magnitude equals the volume of the parallelepiped generated by the three vectors.

Cyclic permutations leave it unchanged:

(34)

Interchanging two vectors reverses the sign.

The Vector Triple Product satisfies

(35)

Equation (35) is often remembered as the BAC-CAB identity, but it is safer to recognize its structural form: the result lies in the plane spanned by B and C.

7 Position and displacement vectors

Choose an origin O.

The position vector of point P is

(36)

In Cartesian coordinates,

(37)

If a second point has position r′, the displacement from the source point r′ to the field point r is

(38)

Its magnitude is

(39)

The corresponding unit vector is

(40)

This source-to-field-point geometry will appear constantly in Coulomb, Biot-Savart, Green-function, multipole, and radiation integrals.

8 A vector field is not one vector

A vector field assigns a vector to each point in space:

(41)

If time dependence is included,

(42)

For example,

(43)

means that every spatial point has its own electric-field vector.

A scalar field instead assigns one scalar value to each point:

(44)

Electric potential is a scalar field; electric field is a vector field.

This difference becomes important in EDL03. A potential map consists of measured scalar values V (x,y) on a grid. From spatial variation of that scalar field we will later infer a vector field E.

9 Coordinate systems as geometric maps

A coordinate system assigns numbers to spatial points.

In Cartesian coordinates,

(45)

labels a point.

In cylindrical coordinates,

(46)

labels the same physical space.

In spherical coordinates,

(47)

provides another labeling.

The coordinate system is not the physical space itself. It is a description adapted to geometry and symmetry.

This is why coordinate choice matters in electrodynamics. A spherical charge distribution is usually easiest in spherical coordinates, a long straight wire in cylindrical coordinates, and a rectangular electrode arrangement in Cartesian coordinates.

10 General orthogonal curvilinear coordinates

Let three coordinates be

(48)

The position vector is a function

(49)

Hold q₂ and q₃ fixed while varying q₁. This traces a coordinate curve.

The tangent to that curve is

(50)

Similarly,

(51)

point along the other coordinate curves.

Define the scale factors

(52)

Then the corresponding unit vectors are

(53)

For an orthogonal coordinate system,

(54)

11 Deriving the differential line element

Differentiate Equation (49):

(55)

Using Equations (52) and (53),

(56)

For orthogonal coordinates,

(57)

The scale factors therefore convert coordinate increments into physical lengths.

This interpretation is crucial. A change dϕ is an angle, not a physical distance. The corresponding physical arc length depends on where the point is located.

12 Differential area elements

Consider a small surface on which q₁ is constant.

The two tangent edge vectors are

(58)

and

(59)

Their cross product gives the oriented area element:

(60)

For a right-handed orthogonal basis,

(61)

Cyclically,

da2	= h3h1 dq3dq1 e2,	(62)
da3	= h1h2 dq1dq2 e3.	(63)

These formulas will later become the geometric foundation of electric and magnetic flux integrals.

13 Differential volume element

The small coordinate cell has edge lengths

(64)

Therefore its differential volume is

(65)

This is the origin of factors such as

(66)

in cylindrical volume elements and

(67)

in spherical volume elements. They are geometric scale factors, not arbitrary memorized corrections.

14 Cartesian coordinates derived from the general construction

The Cartesian position vector is

(68)

Therefore

(69)

(70)

and

(71)

Hence

(72)

Thus

(73)

and

(74)

The Cartesian basis vectors are spatially constant:

(75)

with analogous results for y and z.

This simplicity is special. It does not hold for cylindrical or spherical unit vectors.

15 Cylindrical coordinates

Use cylindrical coordinates

(76)

where s is distance from the z axis, ϕ is azimuthal angle, and z is the usual Cartesian height.

The coordinate transformation is

x	= s cos ϕ,	(77)
y	= s sin ϕ,	(78)
z	= z.	(79)

Therefore

(80)

Differentiate with respect to s:

(81)

Its magnitude is unity, so

(82)

and

(83)

Differentiate Equation (80) with respect to ϕ:

(84)

Its magnitude is

(85)

Therefore

(86)

Finally,

(87)

so

(88)

Thus the cylindrical scale factors are

(89)

Cylindrical line element

Substituting Equation (89) into Equation (56),

(90)

The scalar distance element is

(91)

Cylindrical area elements

For a surface of constant s,

(92)

For a surface of constant ϕ,

(93)

For a surface of constant z,

(94)

Cylindrical volume element

Using Equation (65),

(95)

The factor s appears because an angular step dϕ spans a longer physical distance at larger radius.

16 The cylindrical basis changes with position

Differentiate Equation (83) with respect to ϕ:

(96)

Similarly,

(97)

Therefore a vector field can vary even when its cylindrical components appear numerically constant.

For example,

(98)

has constant radial component F₀, but the direction of the vector changes around the axis.

This basis variation is one of the main reasons divergence and curl formulas contain geometry-dependent terms in curvilinear coordinates.

17 Spherical coordinates

Use spherical coordinates

(99)

with r radial distance, 𝜃 polar angle measured from +z, and ϕ azimuthal angle about the z axis.

The coordinate transformation is

x	= r sin 𝜃 cos ϕ,	(100)
y	= r sin 𝜃 sin ϕ,	(101)
z	= r cos 𝜃.	(102)

Hence

r =	r sin 𝜃 cos ϕx + r sin 𝜃 sin ϕy	(103)
	+ r cos 𝜃 z.	(104)

Differentiate with respect to r:

=	sin 𝜃 cos ϕx + sin 𝜃 sin ϕy	(105)
	+ cos 𝜃 z.	(106)

Its magnitude is unity, so

(107)

and

r =	sin 𝜃 cos ϕx + sin 𝜃 sin ϕy	(108)
	+ cos 𝜃 z.	(109)

Differentiate Equation (104) with respect to 𝜃:

=	r cos 𝜃 cos ϕx + r cos 𝜃 sin ϕy	(110)
	− r sin 𝜃 z.	(111)

Its magnitude is r, so

(112)

and

𝜃 =	cos 𝜃 cos ϕx + cos 𝜃 sin ϕy	(113)
	− sin 𝜃 z.	(114)

Finally,

=	− r sin 𝜃 sin ϕx	(115)
	+ r sin 𝜃 cos ϕy.	(116)

Its magnitude is

(117)

Therefore

(118)

The spherical scale factors are therefore

(119)

Spherical line element

Using Equation (119),

(120)

Thus

(121)

Spherical area elements

At constant r,

(122)

At constant 𝜃,

(123)

At constant ϕ,

(124)

Spherical volume element

The differential volume is

(125)

The factor r² sin 𝜃 follows directly from the three physical edge lengths

(126)

18 Coordinate singularities are descriptive, not necessarily physical

Cylindrical coordinates become singular on the axis

(127)

because the azimuthal angle ϕ becomes undefined there.

Spherical coordinates become singular at

(128)

and along the polar axis where

(129)

A coordinate singularity does not automatically indicate a physical singularity.

For example, a perfectly finite uniform electric field can be expressed in spherical components that have awkward angular behavior near the poles. The underlying physical field may remain smooth even when the coordinate description becomes degenerate.

This distinction will become important when interpreting field singularities near point charges versus harmless singularities introduced by coordinate choice.

19 Electrodynamics examples of symmetry-adapted coordinates

Coordinate systems are most powerful when they match the symmetry of the source.

Point charge

A point charge at the origin produces an electric field whose direction is radial:

(130)

Spherical coordinates expose this symmetry immediately.

Infinite line charge

For an ideal line charge along the z axis,

(131)

Cylindrical coordinates make the radial dependence explicit.

Long straight current

For an ideal current along the z axis, the magnetic field circles the wire:

(132)

Again, cylindrical coordinates align with the field geometry.

Parallel plates

Between large ideal parallel plates with negligible edge effects,

(133)

Cartesian coordinates are natural.

20 Worked example 1: components of a vector

Let

(134)

Its magnitude is

A	=	(135)
	= 13.	(136)

The unit vector in the direction of A is

(137)

If

(138)

then the scalar component of A along B is simply

(139)

21 Worked example 2: source-to-field displacement

Let the source point be

(140)

and the observation point be

(141)

Then

(142)

Its magnitude is

(143)

Therefore

(144)

This exact geometric construction appears in the field integral of a continuous charge distribution.

22 Worked example 3: cylindrical basis at a specific point

Consider the point

(145)

Then

s	= cos 60∘x + sin 60∘y	(146)
	= x + y,	(147)

while

ϕ	= − sin 60∘x + cos 60∘y	(148)
	= −x + y.	(149)

The basis is orthonormal:

(150)

At a different azimuth, the numerical Cartesian components of the cylindrical unit vectors change even though the coordinate names remain s and ϕ.

23 Worked example 4: area of a spherical patch

At fixed radius r, consider a small patch spanning angular increments d𝜃 and dϕ.

The meridional edge length is

(151)

The azimuthal edge length is

(152)

Therefore the patch area is

da	= dl𝜃dlϕ	(153)
	= r2 sin 𝜃 d𝜃 dϕ.	(154)

The corresponding outward vector area is

(155)

This is exactly the area element that will be used in spherical electric-flux calculations.

24 Bridge to the EDL laboratory series

The coordinate geometry developed here is directly measurable.

In EDL03, a potential map may be sampled on a rectangular grid:

(156)

The grid is a Cartesian coordinate description of the physical conducting sheet.

Later experiments may map magnetic fields around a wire or coil. Those measurements often become clearer when replotted against cylindrical radius s or axial coordinate z.

The important experimental lesson is that a coordinate system is part of the data model.

When recording a field measurement, one should specify

• where the sensor is located,
• what coordinate system describes that location,
• which direction the sensor measures,
• and which basis is used to report the measured components.

A Hall probe that reports one scalar channel is not directly measuring an entire vector B. It is measuring a component of B along the probe’s sensitive axis.

Likewise, a potential probe measures a scalar V , not a vector field.

The distinction between field, component, coordinate, and instrument orientation is therefore experimentally important, not merely mathematical bookkeeping.

25 Coordinate-choice workflow

For an electrodynamics problem:

1. Identify the physical symmetry of the source and boundaries.
2. Decide which coordinate surfaces naturally follow that geometry.
3. Write the position vector in those coordinates.
4. Derive or recall the basis vectors and scale factors.
5. Construct the needed differential line, area, or volume element.
6. Express the field components in the local basis.
7. Only then perform the integral or differential operation.

This procedure reduces memorization and makes later formulas easier to reconstruct.

26 Common mistakes

1. Treating a vector and its components as the same object.
2. Forgetting that curvilinear basis vectors vary with position.
3. Using dϕ as though it were a physical length rather than an angle.
4. Omitting the factor s from cylindrical area or volume elements.
5. Omitting r² sin 𝜃 from spherical volume elements.
6. Measuring spherical polar angle 𝜃 from the xy plane rather than from the +z axis when using the Griffiths convention.
7. Confusing the source coordinate r′ with the field coordinate r.
8. Interpreting a coordinate singularity as automatically implying a physical singularity.
9. Using a convenient coordinate system that does not respect the symmetry and then hiding the physics under unnecessary algebra.
10. Differentiating curvilinear vector components while silently treating the basis as constant.

27 Practice exercises

1. For A = 2x − 3y + 6z, find A and A.
2. For A = x + 2y and B = 3x −y, compute A ⋅ B and the angle between them.
3. For the same vectors, compute A × B and interpret its direction.
4. Prove that A ⋅ (B × C) changes sign when any two vectors are exchanged.
5. Starting from Equation (80), rederive s and ϕ.
6. Show explicitly that s ⋅ϕ = 0.
7. Starting from Equation (104), derive h_r, h_𝜃, and h_ϕ.
8. Derive the cylindrical volume element from the product of physical edge lengths.
9. Derive the spherical volume element from the three spherical scale factors.
10. Explain why r sin 𝜃 dϕ becomes small near the poles even when dϕ is fixed.
11. Write the Cartesian vector x in the cylindrical basis at arbitrary ϕ.
12. Show that a radial cylindrical vector field F₀s changes direction with ϕ even though its radial component is constant.
13. For a point charge at the origin, explain why spherical coordinates are symmetry adapted before using any field equation.
14. For an infinite straight wire along z, explain why ϕ is the natural magnetic-field direction.
15. Design a coordinate table for an EDL magnetic-field mapping experiment around a straight current-carrying wire.

28 Companion files

Two compact reference files accompany this article in the PhysicsLibrary filebox.

• Download the ED02 coordinate-geometry summary
• Download the ED02 vector-products summary

These are intended as quick references while solving later ED and EDL problems.

29 Summary

A vector is a geometric object whose components depend on the chosen basis:

(157)

The dot product measures parallel alignment:

(158)

The cross product measures oriented perpendicular content:

(159)

For general orthogonal coordinates,

(160)

and

(161)

The volume element is

(162)

For cylindrical coordinates,

(163)

while for spherical coordinates,

(164)

These geometric results are the foundation from which ED03 will derive gradient, divergence, curl, and the Laplacian rather than treating those operators as formulas to memorize.

References

References