Feynman’s Lectures on Physics 11-4: Vectors – The Mathematics of Symmetry
In the preceding sections of The Feynman Lectures on Physics, Richard Feynman has been developing an argument about the relationship between physical laws and the coordinate systems we use to describe the world. In Chapter 11-1, he introduces the idea that the laws of physics should not depend on where we choose to place the origin of our coordinate system. In 11-2, he considers translations: moving the coordinate axes from one position to another. In 11-3, he examines rotations, demonstrating that the laws of mechanics should remain unchanged when we turn our coordinate system.
This is the purpose of vector analysis.
However, Feynman’s discussion is about considerably more than a convenient mathematical notation. It is an exploration of how we can construct mathematical descriptions of physical reality that preserve the underlying symmetry of nature.
From symmetry to mathematical machinery
Feynman begins by reminding us that the laws of physics, as far as we understand them, possess invariance under translations and rotations of the coordinate axes.
This is a profound observation. If we conduct an experiment in a laboratory, we should obtain the same physical relationships whether we describe the apparatus using one set of Cartesian axes or another. We might rotate the entire coordinate system through some arbitrary angle, but the underlying physics has not changed.
The difficulty is that conventional coordinate-based mathematics can make this invariance rather laborious to demonstrate. When we rotate the axes, the numerical values of the coordinates change. Consequently, equations written in terms of those coordinates must be transformed accordingly.
Feynman has already demonstrated this process, but recognises that repeatedly performing such calculations is inefficient.
His solution is to introduce a mathematical object that can represent a physical quantity independently of the particular coordinate system used to describe it.
This is the vector.
The important distinction is that vector analysis does not simply provide a way of drawing arrows or resolving forces. It provides a mathematical framework in which the symmetry of physical relationships is built into the notation itself.
An equation expressed correctly in vector form automatically retains its meaning when the coordinate axes are rotated.
This is an enormous simplification, but it is also conceptually important: the mathematics begins to distinguish between the physical reality being described and the arbitrary conventions we use to describe it.
Scalars and vectors: two fundamentally different kinds of quantity
Feynman introduces vectors by making a distinction between two broad categories of physical quantities.
The first are scalars, which he also calls ordinary or undirected quantities.
His deliberately uncomplicated example is the number of potatoes in a sack. We might have twelve potatoes, but there is no meaningful direction associated with that number. Temperature is another example. A room might be at 20°C, but the temperature does not point towards the north, south, upwards or downwards.
A scalar is therefore completely described by a single numerical value, together with its appropriate units.
The second category comprises vectors: quantities that possess both magnitude and direction.
Feynman uses velocity, momentum, force and displacement as examples. If somebody walks five metres, we know the distance travelled, but we do not know where they went. To describe their displacement, we also need a direction.
This distinction is particularly important in mechanics.
A car travelling at 20 metres per second has a speed of 20 metres per second. However, its velocity requires us to specify its direction of travel as well. Two cars can have identical speeds but different velocities if they are travelling in opposite directions.
Similarly, a force of 10 newtons is not fully specified until we know the direction in which it acts.
The distinction between scalar and vector quantities is therefore not merely a mathematical convention. It reflects a fundamental difference in the physical information required to describe different properties of the world.
It is worth noting that Feynman is deliberately beginning with just two categories. Physics contains other mathematical objects, including tensors, which become necessary when describing quantities with more complicated transformation properties. But scalars and vectors provide the essential starting point.
A vector is three numbers – but not quite
One of the most interesting parts of Feynman’s explanation is his description of a vector as three numbers.
Consider a point P in three-dimensional space. Relative to an origin, we can describe its position using three Cartesian coordinates: x, y and z.
Feynman introduces a single mathematical symbol, r, to represent these three numbers.
This is a slightly unusual idea. We are accustomed to mathematical symbols representing individual numbers, but here one symbol represents a collection of three numerical components.
The crucial qualification is that these three numbers depend upon the coordinate system.
If we rotate our axes, the coordinates of P change. The original values of x, y and z become a different set of coordinates, which we might call x′, y′ and z′.
Yet the physical point P has not moved.
This is the conceptual heart of Feynman’s discussion.
The vector r represents the position of the point, not simply the particular numbers used to describe it. Those numbers are its components in a chosen coordinate system.
Feynman therefore uses the same symbol, r, regardless of which coordinate system we employ.
This is a subtle but powerful distinction. We are separating the physical object from its mathematical representation.
It is rather like describing a geographical location using latitude and longitude or using a different system of coordinates. The numerical description changes, but the location itself does not.
The same principle applies to displacement, velocity, force and other vector quantities.
A vector is therefore not simply a collection of three numbers. It is a physical or mathematical quantity whose components change in a particular, well-defined way when the coordinate system changes.
That qualification is essential.
The transformation properties define a vector
Feynman now makes an important conceptual leap.
Suppose we have some physical quantity represented by three numbers. When we rotate our coordinate axes, those three numbers change.
How do we know whether this quantity is a vector?
Feynman’s answer is that it must transform in exactly the same way as the components of a displacement in space.
This is a much more rigorous definition than simply saying that a vector has magnitude and direction.
It is the transformation behaviour that matters.
For example, consider a force acting on an object. We can describe its components along three mutually perpendicular axes. If we rotate those axes, the components of the force change.
However, they must change according to precisely the same geometrical rules as the components of a displacement.
This allows us to identify force as a vector.
The same applies to velocity, acceleration and momentum.
There is a deeper mathematical idea here. A vector is not defined by the appearance of its components, but by how those components behave under a change of coordinates.
This is particularly useful because it gives us a way of identifying vectors without needing to know what physical phenomenon they represent.
Any three-component quantity that obeys the appropriate transformation law qualifies.
This approach also points towards the more sophisticated mathematical language of tensors, in which physical quantities are classified according to their transformation properties under changes of coordinates.
Feynman is introducing the essential philosophy behind that subject without burdening the reader with its formalism.
Why vector equations are independent of coordinate systems
Feynman illustrates the advantage of this approach using a simple relationship between two vectors.
Imagine that a force vector is proportional to a displacement vector, with some constant of proportionality.
The important feature of expressing this relationship as a vector equation is that it represents three simultaneous scalar equations.
In one coordinate system, the x-components must correspond, the y-components must correspond, and the z-components must correspond.
If we rotate the coordinate system, all six component values change, but the relationship between the vectors remains intact.
The equation itself does not need to be rewritten.
This is the practical advantage of vector notation.
An ordinary coordinate-based equation can become cumbersome when expressed in a new coordinate system. A vector equation, however, retains its form because both sides transform according to the same rules.
There is an important qualification here: the equation must be a valid vector equation in the first place. It is not enough simply to place arrows over quantities or group three numbers together. Both sides must represent objects with compatible transformation properties.
When this condition is satisfied, the equation expresses a physical relationship that is independent of the arbitrary orientation of our coordinate axes.
This is why vector notation is so fundamental to physics.
It allows us to distinguish between a mathematical statement that happens to work in one particular coordinate system and a physical law that is valid in every appropriately related coordinate system.
Why can we represent vectors as arrows?
Having established the mathematical definition, Feynman turns to a more familiar visual representation.
We routinely draw vectors as arrows.
A force acting on an object might be represented by an arrow pointing in the direction of the force, with its length proportional to the magnitude.
But why is this legitimate?
Feynman’s explanation returns to his original definition: vectors transform in the same way as displacements in space.
A displacement can naturally be represented by an arrow. It has a direction and a length. Since a force has the same transformation properties, we can represent it in the same way.
We simply choose an appropriate scale.
For example, we might decide that one centimetre on a diagram represents one newton. A force of five newtons would then be represented by an arrow five centimetres long.
This is more than a convenient illustration. It reflects the mathematical equivalence of the geometrical behaviour of force and displacement.
Once the vector has been drawn, we no longer necessarily need to show the coordinate axes. The arrow contains the essential information about its magnitude and direction.
If we subsequently want to calculate its components relative to some particular set of axes, we can do so using elementary geometry.
This is an important transition from algebraic manipulation to geometrical reasoning.
Instead of repeatedly transforming three numerical components, we can work directly with the geometrical object they represent.
The vector diagram allows us to visualise the physical relationship without committing ourselves to a particular coordinate system.
The deeper significance: physical reality versus mathematical description
There is a particularly interesting philosophical point running through this section.
Feynman observes that the idea of a single physical object being represented by different sets of coordinates depends upon our intuition that the object exists independently of the coordinate system.
The coordinates are not the object itself. They are a means of describing it.
This distinction is fundamental to modern physics.
When we describe the position of a planet, the velocity of a particle or the force exerted by a magnetic field, we are using mathematical quantities to represent aspects of physical reality. Our choice of axes is a matter of convenience, not a property of the physical system.
A coordinate system can be moved, rotated or replaced altogether without changing the underlying physical situation.
The mathematics must therefore accommodate this distinction.
Vectors provide one of the simplest and most powerful ways of doing so.
There is also a broader connection to the principle of covariance: the idea that physical relationships should be expressible in forms that retain their validity under appropriate transformations of coordinates.
This becomes particularly important in more advanced physics. In electromagnetism, for example, electric and magnetic fields are represented using vector quantities, while in general relativity, the mathematical description of spacetime requires the more general framework of tensors.
The underlying motivation is similar: physical laws should not depend upon arbitrary choices made in our mathematical description.
Feynman is therefore introducing a mathematical language that will become indispensable throughout the rest of physics.
The significance of Feynman’s teaching method
As with the preceding sections, Feynman’s approach is notable for the way he introduces mathematical ideas through physical reasoning.
He does not begin by presenting the formal axioms of vector spaces, nor does he introduce the machinery of matrix transformations. Instead, he starts with something physically intuitive: a displacement in space.
He then asks what happens when we change the coordinate system used to describe that displacement.
From this simple observation, he develops a definition of a vector based on its transformation properties.
This is an effective teaching strategy because it gives the mathematical concept a physical purpose before introducing its formal structure.
It also helps avoid a common misunderstanding among students: that vectors are simply quantities that have magnitude and direction.
That description is useful, but incomplete. The deeper definition concerns how the components of a vector transform when the coordinate system changes.
Feynman also makes a deliberate effort to connect the abstract mathematics with practical diagrams. The arrow representation is introduced not as an arbitrary convention but as a direct consequence of the mathematical properties of vectors.
His treatment is economical, but the ideas are far-reaching.
Conclusion: a mathematical language built around the symmetry of nature
Chapter 11-4 is a relatively short introduction to vectors, but its significance extends well beyond the immediate subject.
Feynman has spent the preceding sections establishing that the laws of physics should remain unchanged when we translate or rotate our coordinate systems. He now introduces vectors as the mathematical machinery that allows us to express those laws efficiently.
The distinction between scalars and vectors provides the starting point. The more important development is the recognition that vectors are defined by their transformation properties, allowing physical quantities to be represented independently of the particular axes used to describe them.
The arrow diagram is therefore not merely a convenient way of drawing forces. It is a geometrical representation of a mathematical object whose physical meaning survives a change of coordinates.
Perhaps the most important lesson is that mathematics should distinguish between the physical world and the conventions we use to describe it.
A coordinate system is something we impose upon space. A vector is a way of describing a physical quantity without allowing that arbitrary choice to dictate the form of the laws governing it.
Feynman is showing us that symmetry is not simply an interesting property of physical laws. It can be used to construct the very mathematical language in which those laws are expressed.
That is the real achievement of vector analysis: it makes the invariance of physical relationships almost automatic, allowing us to concentrate on the physics rather than repeatedly proving that our choice of coordinates has not changed it.
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