So, what is a vector?

In my first year of undergraduate physics, one of my professors walked into the lecture hall, placed his notes on the desk, looked around the room, and asked a question that, at first, seemed almost too easy.

“What is a vector?”

I immediately raised my hand.

“A vector is a quantity that has both magnitude and direction.”

He smiled.

It wasn’t the kind of smile that tells you you’ve answered correctly, nor was it the kind that says you’ve made a mistake. It was the smile of someone who knows there’s a much more interesting answer waiting just around the corner. At the time, I didn’t think much of it. I had given the definition that every mathematics and physics student learns in high school. Surely that was the answer.

After a short pause, he looked at the class and said, “That’s how we often describe vectors. But it isn’t what a vector really is.”

I remember sitting there wondering what he meant. If a vector isn’t simply something with a magnitude and a direction, then what exactly is it? Every example I had ever seen was an arrow. Every textbook drew arrows. Every problem involved arrows. Wasn’t that enough?

Instead of answering the question directly, he told us a story.

He said that mathematics rarely invents new objects for the sake of inventing them. New mathematical ideas almost always appear because the old ones stop being sufficient. People encounter a problem, struggle with it using the mathematics they already have, and eventually realize that their language is no longer rich enough to describe what they are seeing. The new mathematics is born not from imagination, but from necessity.

He asked us to think about the numbers we learn as children. We begin with the natural numbers: $1, 2, 3, \ldots$. For a while, they seem to answer every question we ask. If there are three apples on the table and someone gives you two more, you have five apples. If there are ten students in a classroom and four leave, six remain. The arithmetic is simple, and the natural numbers seem perfectly adequate.

Eventually, however, we encounter questions they cannot answer. What is $3-5$? Within the natural numbers, the expression has no answer. It isn’t that the calculation is wrong; rather, our number system is too limited to express the result. Instead of declaring the question meaningless, mathematics extends the number system. The integers are introduced, and suddenly the answer $-2$ makes perfect sense.

The same story repeats itself. Rational numbers appear because integers cannot always describe parts of a whole. Real numbers appear because rational numbers cannot represent lengths such as $\sqrt{2}$.¹The ancient Greek discovery that the diagonal of a unit square cannot be expressed as a ratio of two integers was one of the earliest demonstrations that the rational numbers are incomplete. Complex numbers appear because some perfectly reasonable equations, such as $x^2+1=0$, simply have no real solutions. Every time mathematics grows, it grows because reality, or mathematics itself, demands a richer language.

“Vectors,” my professor said, “were born for exactly the same reason.”

At first, that statement surprised me. I had always treated vectors as though they were just another topic in geometry, sitting alongside triangles, circles, and coordinate systems. It had never occurred to me that they were invented to solve a problem. But once he said it, the obvious question became unavoidable.

What problem?

To answer that question, we have to forget everything we think we know about vectors for a little while. Forget the arrows. Forget the coordinate axes. Forget the ordered pairs like $(3,4)$. Those are all ways of representing vectors, but they are not where the idea begins.

Instead, let us go back to a time before the word “vector” even existed and ask a much simpler question: what kinds of quantities can ordinary numbers actually describe?

Some quantities surrender completely to a single number. The temperature of this room. The mass of an electron. The time on the clock at the back of the lecture hall. Tell me the number and you have told me everything; there is nothing left over. And these numbers have a second property, so obvious that it hides. Everyone in the room agrees on them, no matter which way they happen to be facing. If I turn my chair forty degrees, the room does not get warmer. Quantities like this are called scalars, and for most of history they were simply called quantities, because nobody imagined there could be another kind.

Then try this one. I leave my house and walk five kilometers. Where am I?

You cannot say. I could be five kilometers north, five kilometers southeast, anywhere on an entire circle of possibilities. The number five is not wrong. It is incomplete. A displacement has a size, but the size does not exhaust it; something is left over after the number has done its work, and that something is which way. The same incompleteness infects velocity, force, acceleration, the pull of a magnet on a compass needle. By the time people were seriously composing forces, adding one push to another and asking for the net effect, the problem could no longer be postponed. A single number per quantity was a language that had run out.²The parallelogram rule for composing forces was stated clearly by Simon Stevin in 1586 and appears as a corollary in Newton’s Principia, so the geometric content of vector addition is centuries older than the word. The word itself comes from William Rowan Hamilton in the 1840s, from the Latin vehere, to carry, as part of his quaternions; the modern, stripped-down vector analysis was carved out of the quaternion system by Gibbs and Heaviside toward the end of the nineteenth century. It is a pleasing echo of the professor’s story: Hamilton was searching for a three-dimensional analogue of the complex numbers, which had themselves been invented when the real numbers ran out.

The tempting fix is to use more numbers. Lay down two reference directions, east and north, and report the walk as a pair: three kilometers east, four kilometers north. Write it $(3, 4)$. Two numbers instead of one, the ambiguity is gone, and you know exactly where I ended up. For a while this feels like the whole answer. It is the answer your textbook’s ordered pairs are quietly assuming.

But a friend of mine surveys the same walk, and she has laid her reference directions differently. Her “east” points along the road outside her house, which runs at an angle to mine. She measures carefully and reports the same walk as roughly $(4.6, 2.0)$. Same starting point, same ending point, same physical walk through the same physical mud. Different numbers.

If a vector were a pair of numbers, we would now have a contradiction: one object equal to two different things. The walk cannot be $(3,4)$ and also $(4.6, 2.0)$. So the walk is not the pair of numbers. The pair of numbers is a description of the walk in a particular language, and the language is the choice of axes. Her language differs from mine, so her words differ from mine, and the walk itself does not care about either of us.

This is the moment to slow down, because everything that follows grows out of it. The numbers change with the observer; the object does not. If we want the mathematics to describe the object rather than the observer, we have to find out what, in all this shifting of numbers, stays put.

And here the situation is far better than it first appears, because the two descriptions are not unrelated. If I know the single angle $\theta$ between my axes and hers, I can compute her numbers from mine, exactly, without ever leaving my desk. Suppose my components are $v_x$ and $v_y$, and write $|v|$ for the length of the displacement and $\varphi$ for the angle it makes with my east axis, so that $v_x = |v|\cos\varphi$ and $v_y = |v|\sin\varphi$. Her east axis sits at angle $\theta$ from mine, which means the same displacement makes angle $\varphi – \theta$ with her axis, and her components are $|v|\cos(\varphi-\theta)$ and $|v|\sin(\varphi-\theta)$. Expanding the angle differences³The two identities used here are $\cos(\varphi-\theta) = \cos\varphi\cos\theta + \sin\varphi\sin\theta$ and $\sin(\varphi-\theta) = \sin\varphi\cos\theta – \cos\varphi\sin\theta$. They are the whole trigonometric input of this article; everything else is substitution. and substituting $v_x$ and $v_y$ back in gives

$$ v’_x = v_x\cos\theta + v_y\sin\theta, \qquad v’_y = -\,v_x\sin\theta + v_y\cos\theta. $$

Check it on the walk. My components were $(3,4)$, and suppose her axes are rotated by $\theta = 30^\circ$, so $\cos\theta \approx 0.866$ and $\sin\theta = 0.5$. Then $v’_x = 3(0.866) + 4(0.5) \approx 4.60$ and $v’_y = -3(0.5) + 4(0.866) \approx 1.96$. Those are her numbers, the ones she measured in the field, recovered from mine by pure computation. The two languages are connected by a dictionary, and the dictionary is this pair of formulas.⁴A bookkeeping note: throughout, the walk is held fixed and the axes rotate, which is called a passive transformation. One can instead hold the axes fixed and rotate the object by $\theta$, an active transformation; the formulas are the same with $\theta$ replaced by $-\theta$. Both conventions are common, which is why signs in other books may differ from mine.

But where does the dictionary itself come from? The trigonometric derivation shows that it is correct. It does not yet show that it is forced. So let me derive it a second time, from a different starting point, because the second derivation is the one that says what a vector actually is. Introduce symbols for my two reference directions themselves: let $e_x$ and $e_y$ be the unit displacements, one kilometer east and one kilometer north. Then my report $(3,4)$ was shorthand for an equation about objects,

$$ v = 3\,e_x + 4\,e_y, $$

three copies of the first reference displacement followed by four of the second, and in general $v = v_x\,e_x + v_y\,e_y$. My friend has her own reference displacements, $e’_x$ and $e’_y$. Since hers are ordinary displacements like any others, I can describe them in my language. Her first axis points at angle $\theta$ from mine, so

$$ e’_x = \cos\theta\, e_x + \sin\theta\, e_y, \qquad e’_y = -\sin\theta\, e_x + \cos\theta\, e_y. $$

The crucial demand comes next, and it is a physical demand, not a mathematical one: she and I are describing the same walk. Whatever components she uses must reassemble the identical object,

$$ v_x\, e_x + v_y\, e_y = v’_x\, e’_x + v’_y\, e’_y. $$

Substitute her basis vectors and collect my $e_x$ and $e_y$:

$$ v_x\, e_x + v_y\, e_y = (v’_x\cos\theta – v’_y\sin\theta)\, e_x + (v’_x\sin\theta + v’_y\cos\theta)\, e_y. $$

Two walks reported in the same language, east part and north part, are the same walk only if the east parts agree and the north parts agree.⁵I am borrowing a fact here: that a displacement in the plane determines its east part and north part uniquely. Geometrically this is clear enough. In the general setting it is exactly the uniqueness of representation in a basis, which I prove properly once bases are defined below, at which point this step stops being a loan. So $v_x = v’_x\cos\theta – v’_y\sin\theta$ and $v_y = v’_x\sin\theta + v’_y\cos\theta$. Solve the pair for the primed components, multiplying the first by $\cos\theta$ and the second by $\sin\theta$, adding, and using $\cos^2\theta + \sin^2\theta = 1$, and out comes precisely the dictionary we derived with angles:

$$ v’_x = v_x\cos\theta + v_y\sin\theta, \qquad v’_y = -\,v_x\sin\theta + v_y\cos\theta. $$

Nothing about the law was a convention. Once you grant that the two of us describe one object, the law is the only possibility.

Two rules are now on the table, one telling how the basis vectors translate and one telling how the components translate, and the relationship between them is the point this whole article has been walking toward. They are inverses of each other. For pure rotations the inverse of the translation array happens to coincide with its transpose, which disguises the opposition, so let me expose it with a cruder change of language. Suppose my friend does not rotate her axes at all but merely adopts a longer measuring stick, so that her reference displacement is twice mine, $e’_x = 2\,e_x$. The walk is the walk: $v = v_x\,e_x = v’_x\,e’_x = 2\,v’_x\,e_x$, so $v’_x = v_x/2$. Her basis vector doubled and her component halved. The two always pull in opposite directions, and what their pulling multiplies out to, the object, sits perfectly still. A vector is a tug of war between basis and components that ends, in every language, in exactly the same draw.⁶This opposition has standard names: components that transform inversely to the basis are called contravariant, and objects that transform along with the basis, as the gradient does, are called covariant. The raised and lowered indices of the previous post are precisely this bookkeeping, $x^\mu$ upstairs for contravariant components and $\partial_\mu$ downstairs for covariant objects, and the convention exists so that a paired upper and lower index always assembles a language-independent quantity.

Now ask what the dictionary preserves. Take her components, square them, and add:

$$ v_x’^2 + v_y’^2 = (v_x\cos\theta + v_y\sin\theta)^2 + (v_y\cos\theta – v_x\sin\theta)^2. $$

Multiply everything out. The cross terms $2v_xv_y\cos\theta\sin\theta$ appear once with a plus sign and once with a minus sign, and cancel. What remains is $v_x^2$ and $v_y^2$, each multiplied by $\cos^2\theta + \sin^2\theta$, which is $1$. So

$$ v_x’^2 + v_y’^2 = v_x^2 + v_y^2. $$

On the numbers: $3^2 + 4^2 = 25$, and $4.60^2 + 1.96^2 = 21.2 + 3.8 = 25.0$. Twenty-five both times. The individual components shifted, but this particular combination of them refused to move, and it refuses for every angle $\theta$, not just thirty degrees. Its square root is the length of the displacement. So the length is not part of the description; it is a fact about the object, the same in every language. Direction behaves more subtly: the angle my walk makes with my east axis is $\varphi$, while the angle it makes with hers is $\varphi – \theta$, so “the direction” as an angle relative to axes is language-dependent. But take two walks and ask for the angle between them. Both their angles shift by the same $\theta$, and the difference survives untouched. Lengths and mutual angles belong to the objects. Components and axis angles belong to the description.

You can feel the high-school definition being rebuilt on better foundations. “Magnitude and direction” was gesturing at exactly this: the invariant content, the part of the description that all observers share. It just said it sloppily, as though magnitude and direction were the definition rather than the residue that survives translation.

So here is a first serious answer to the professor’s question, the one a physicist tends to give. A vector is not a list of numbers. A vector is a quantity whose lists of numbers, written down by different observers, are tied to one another by exactly the dictionary above. In two dimensions the dictionary is the pair of formulas we derived; in $n$ dimensions it takes the form

$$ v’_i = \sum_{j} R_{ij}\, v_j, $$

where the array $R_{ij}$ encodes the relative orientation of the two frames.⁷For rotations, $R$ is an orthogonal matrix: its rows are the new axis directions written in the old language, it satisfies $\sum_i R_{ij}R_{ik} = \delta_{jk}$, where $\delta_{jk}$ equals $1$ when $j=k$ and $0$ otherwise. This orthogonality is forced rather than assumed: demand that $\sum_i v_i’^2 = \sum_j v_j^2$ for every vector, substitute the transformation law to get $\sum_{j,k}\left(\sum_i R_{ij}R_{ik}\right)v_j v_k$, and the only way the result can equal $\sum_j v_j^2$ for all choices of the $v_j$ is for the bracketed sum to be $\delta_{jk}$. The two-dimensional cancellation we did by hand is the $n=2$ case. Requiring in addition that the frame is not mirror-reflected fixes $\det R = +1$. Physicists compress this into a slogan: a vector is anything that transforms like a vector. Said quickly, it sounds circular. It is not. It is a behavioral definition, like defining a key as anything that opens the lock. It does not tell you what the object is made of; it tells you how to test whether a given candidate qualifies.

One consequence of the law deserves its own sentence, because the whole of mechanics leans on it. The dictionary is linear: it never squares a component or multiplies two of them together. So if $u$ and $v$ both pass the test, their sum passes automatically,

$$ (u+v)’_i = \sum_{j} R_{ij}\,(u_j + v_j) = u’_i + v’_i, $$

and likewise any multiple $a\,u$. Sums of vectors are vectors, and rescaled vectors are vectors. That is the license behind composing forces at all: add two forces tip to tail and the result is guaranteed to be a quantity that every observer’s dictionary handles correctly.

The law also has to pass a consistency requirement involving three observers instead of two, and it is worth checking rather than trusting. Bring in a third surveyor whose axes sit at angle $\theta_2$ from my friend’s, and therefore at $\theta_1 + \theta_2$ from mine. There are now two ways to compute his components from mine: translate directly, using the dictionary with angle $\theta_1 + \theta_2$, or translate in two hops, first into her frame with $\theta_1$ and then onward with $\theta_2$. If the two routes disagreed, the whole scheme would collapse; a component would depend on which intermediate observers you consulted along the way. So take the two-hop route and compute, substituting her components into his dictionary:

$$ v”_x = v’_x\cos\theta_2 + v’_y\sin\theta_2 = v_x\left(\cos\theta_1\cos\theta_2 – \sin\theta_1\sin\theta_2\right) + v_y\left(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2\right). $$

The parentheses are the angle-addition identities run in reverse, the subtraction identities from the first derivation with $\theta$ replaced by $-\theta$, and they collapse:

$$ v”_x = v_x\cos(\theta_1+\theta_2) + v_y\sin(\theta_1+\theta_2). $$

That is exactly the direct dictionary, and the $y$ component checks the same way. Two hops equal one hop. The dictionaries between all possible observers mesh into a single consistent system,⁸In modern language: rotations compose to give rotations, every rotation has an inverse, and doing nothing counts as a rotation, so the collection of all rotations forms a group, written $SO(2)$ in the plane and $SO(n)$ in $n$ dimensions. “Transforms like a vector” is, said fully, a statement about how an object responds to this entire group at once, and group theory is the systematic study of such consistency requirements. It becomes one of the load-bearing structures of modern physics. and a vector is an object that speaks every one of those languages at once, coherently.

And the test has teeth. Take three perfectly good numbers measured at a point in this room: the temperature, the pressure, and the humidity. Stack them into a triple, $(T, p, h)$. Three numbers, arranged in a column; you could even draw the triple as an arrow in an abstract space if you felt like it. Now rotate your axes. Nothing happens. The temperature does not care which way you face, and neither do the other two; each is a scalar, so the triple you write down after the rotation is the same triple you wrote before. But the components of a genuine vector mix under rotation, each new component a blend of all the old ones, as our formulas demand. This triple refuses to mix. It fails the test. Three numbers in a column are not a vector, any more than three people in an elevator are a family.

The demolition works from the other side too, and this one surprised me more. There are things in physics that possess a perfectly good magnitude and a perfectly good direction and are still not vectors. A finite rotation is one: it has an axis, which is a direction, and an angle, which is a magnitude. Try to add two of them. Hold a book flat in front of you, rotate it $90^\circ$ about the vertical axis and then $90^\circ$ about the horizontal axis pointing away from you, and note where it ends up. Restore it, and apply the same two rotations in the opposite order. The book lands in a visibly different position. But vector addition does not care about order; $a + b$ and $b + a$ are the same walk. Finite rotations, magnitude and direction and all, break the rules of the game.⁹The failure is a finite-angle effect. Rotations through very small angles do commute, up to corrections of second order in the angles, and this is why angular velocity, which is built from infinitesimal rotations, is a legitimate vector while “the rotation itself” is not. There is a further subtlety I am deliberately postponing: quantities like angular velocity behave correctly under rotations but pick up an extra sign under mirror reflection, which is why they are called axial vectors or pseudovectors. That distinction deserves its own discussion. So magnitude-and-direction is neither sufficient to make something a vector, as the rotations show, nor, as we are about to see, even necessary.

Because now the mathematician takes a turn, and the mathematician’s move is more radical than anything so far. Forget the transformation law for a moment and ask an operational question: in all our dealings with displacements, what did we ever actually do with them? Two things. We added them: walk $u$ and then walk $v$, and the net effect is a single walk written $u + v$, running from the start of the first to the tip of the second. And we scaled them: twice the walk, half the walk, the reverse of the walk. That is the complete inventory. Every construction we performed was built from adding and scaling. And the addition has a property worth drawing rather than asserting: it does not care about order.

Both routes around the parallelogram, $u$ then $v$ along the lower flank, $v$ then $u$ along the upper, arrive at the same corner, so $u + v = v + u$. For displacements, the commutativity that finite rotations just failed is a closed quadrilateral you can check with a ruler.

So the mathematician defines: a vector space is any collection of objects that can be added to each other and multiplied by numbers, provided those two operations obey the familiar rules of arithmetic, and a vector is nothing more and nothing less than a member of such a collection.¹⁰The rules, stated compactly: addition is commutative and associative, there is a zero element, every element has a negative, scaling distributes over both vector addition and number addition, scaling is compatible with multiplication of numbers, and scaling by $1$ does nothing. Eight axioms in all. The “numbers” doing the scaling can be the real numbers or the complex numbers; quantum mechanics is built on complex vector spaces, and everything in this article goes through in either case. Read the definition twice and you will see what it does not say. Nothing about arrows. Nothing about magnitude. Nothing about direction. Nothing, even, about space. The definition kept the algebra of displacements and threw away the picture.

And the rules are not decoration. They prove things. Take the smallest possible example: what is a vector scaled by the number zero? For arrows the answer is obvious, but the definition never mentions arrows, so the answer must be squeezed out of the rules alone. Since $0 = 0 + 0$ as numbers, distributivity gives

$$ 0\,v = (0+0)\,v = 0\,v + 0\,v. $$

Add to both sides the negative of $0\,v$, which the rules guarantee exists. The left side becomes the zero element, the right side becomes $0\,v$, and so

$$ 0\,v = \mathbf{0}. $$

Scaling anything by zero gives the zero vector, in every vector space there is or ever will be, polynomials and quantum states included, and we proved it without knowing what the vectors are made of. That is what an axiomatic definition buys: theorems with unlimited jurisdiction.

The payoff for that austerity is enormous, because the definition now admits members the picture could never have suggested. Take polynomials. Add two of them,

$$ (1 + x^2) + (3x – x^2) = 1 + 3x, \qquad 2\,(1+x^2) = 2 + 2x^2, $$

and you can check, patiently, that all eight rules hold. Polynomials form a vector space. So the polynomial $1 + x^2$ is a vector, and here is the question that finishes off the high-school definition for good: what is its direction? The question is not hard. It is empty. Nothing in the space of polynomials points anywhere, and yet the addition and the scaling work flawlessly. Direction was never the essence. It was a feature of one famous example.

The same is true of functions at large, and this is where the abstraction starts paying rent in physics. In the previous post we derived the wave equation, $\Box\phi = 0$. Suppose $\phi_1$ and $\phi_2$ are both solutions. Then for any numbers $a$ and $b$,

$$ \Box\,(a\,\phi_1 + b\,\phi_2) = a\,\Box\phi_1 + b\,\Box\phi_2 = 0, $$

so the combination is a solution too. The solutions of the wave equation form a vector space. You have known this fact for years under a different name: superposition. When two ripples cross a pond and pass through each other, the water is performing vector addition in an infinite-dimensional function space. The principle of superposition is the statement that the solution set is a vector space, and it holds for exactly as long as the governing equation is linear.

At this point the two definitions, the physicist’s and the mathematician’s, look like they belong to different books. One is about dictionaries between observers; the other is about the arithmetic of abstract sets. The bridge between them is the idea of a basis, which has in fact been operating informally ever since the reference displacements $e_x$ and $e_y$ first appeared. It deserves to be defined with care rather than gestured at. A set of vectors $e_1, e_2, \ldots, e_n$ is called a basis for the space if it does two jobs at once. First, it must span: every vector $v$ in the space can be written as some combination

$$ v = v_1\, e_1 + v_2\, e_2 + \cdots + v_n\, e_n. $$

Second, it must be linearly independent, which means the reference vectors carry no internal redundancy: the only combination of them that produces the zero vector is the trivial one,

$$ c_1\, e_1 + c_2\, e_2 + \cdots + c_n\, e_n = \mathbf{0} \quad\Longrightarrow\quad c_1 = c_2 = \cdots = c_n = 0. $$

Spanning guarantees that a description exists. Independence guarantees something better, that the description is unique, and the proof takes three lines. Suppose some vector had two descriptions, $v = a_1 e_1 + \cdots + a_n e_n$ and also $v = b_1 e_1 + \cdots + b_n e_n$. Subtract them:

$$ (a_1 – b_1)\,e_1 + (a_2 – b_2)\,e_2 + \cdots + (a_n – b_n)\,e_n = \mathbf{0}. $$

Independence forces every coefficient to vanish, so $a_i = b_i$ for each $i$, and the two descriptions were the same description all along. This also repays the loan from earlier: matching the east parts and north parts of two equal walks was exactly this argument, run in the plane. The numbers $v_1, \ldots, v_n$ are the components of $v$, and now we can say precisely what they are: not properties of $v$ alone, but properties of the pair, the vector together with the chosen basis. Nor is any of this confined to arrows. The monomials $1, x, x^2, x^3, \ldots$ form a basis for the polynomials,¹¹Independence here means: if $c_0 + c_1 x + \cdots + c_n x^n$ is the zero vector of this space, which is the polynomial equal to zero for every value of $x$, then all the $c_i$ vanish. Set $x=0$ and $c_0$ falls. Differentiate once, set $x=0$ again, and $c_1$ falls. Keep differentiating; each coefficient drops in turn. So the monomials carry no redundancy, and the components below are the unique ones. and in that basis the polynomial $1 + 3x$ from a few paragraphs ago has components $(1, 3, 0, 0, \ldots)$: one part constant, three parts $x$, nothing else. What looked like algebra was a coordinate representation all along.¹²A basis is far from unique, but a theorem guarantees that every basis of a given space contains the same number of elements, and that shared number is the dimension of the space. Displacements in a plane have dimension two, displacements in the room have dimension three, and the polynomials and the wave-equation solutions above have infinite dimension, which changes some technical details and none of the conceptual ones. My friend and I were not disagreeing about the walk. We had selected different bases, hers rotated relative to mine, and the transformation law we derived with so much trigonometry is nothing but the recipe for converting components with respect to one basis into components with respect to another. The physicist’s definition is the mathematician’s definition plus one nonnegotiable demand: nature never saw our axes, so no physical statement may depend on which basis we happened to pick. It is the same instinct that made us write the invariant volume $\sqrt{-g}\,d^4x$ in the previous post instead of the coordinate volume $d^4x$. The laws belong to the objects, not to the bookkeeping.

Once you see components as coordinates-with-respect-to-a-basis, a familiar tool from a completely different corner of physics turns out to be this same idea in disguise. Functions on an interval form a vector space, and the sines and cosines can serve as a basis for it. Writing a signal as a Fourier series is choosing that basis and reading off the components; the Fourier coefficients of a sound wave are, in the most literal sense, the components of a vector.¹³To make “component along a basis function” precise you need an inner product on the function space, and the natural one is an integral, $\langle f, g\rangle = \int f(x)\,g(x)\,dx$ over the interval. The sines and cosines are orthogonal with respect to it, which is what makes the coefficients easy to extract, one integral each. Infinite dimensions bring genuine subtleties about convergence that I am stepping over here, but the geometric picture, a function as a vector and its Fourier coefficients as components, is exactly right. Nobody would ask for the magnitude and direction of a violin note. Its vectorhood lies elsewhere.

Which raises the last question honestly: if direction is not essential, what exactly is the status of magnitude? The answer is that magnitude is optional equipment. To speak of lengths you must add a further gadget to the space, an inner product, the familiar dot product being the standard issue:

$$ u \cdot v = u_x v_x + u_y v_y. $$

Run it through the dictionary and watch the cancellation happen, this time in full. Her value of the product is

$$ u’\cdot v’ = (u_x\cos\theta + u_y\sin\theta)(v_x\cos\theta + v_y\sin\theta) + (u_y\cos\theta – u_x\sin\theta)(v_y\cos\theta – v_x\sin\theta). $$

Multiply out all four products. The terms carrying $\cos\theta\sin\theta$ arrive in matched pairs of opposite sign, $u_x v_y$ once with a plus and once with a minus, $u_y v_x$ the same, and annihilate. What survives is $u_x v_x$ and $u_y v_y$, each dressed with a factor of $\cos^2\theta + \sin^2\theta$:

$$ u’\cdot v’ = u_x v_x + u_y v_y = u\cdot v. $$

The dot product of two vectors comes out identical in every rotated frame, and the length computation we did earlier is the special case $u = v$. From the dot product you then recover everything the arrows promised, length as $|v| = \sqrt{v\cdot v}$ and the angle between two vectors through $u\cdot v = |u|\,|v|\cos\alpha$.¹⁴That last formula only defines an angle if the quantity $u\cdot v/(|u||v|)$ actually lies between $-1$ and $1$, and the guarantee is the Cauchy–Schwarz inequality, $(u\cdot v)^2 \le (u\cdot u)(v\cdot v)$. The proof is short enough to give in full. For any number $t$, the vector $u – t\,v$ has a non-negative dot product with itself, so $0 \le (u – t\,v)\cdot(u – t\,v) = u\cdot u – 2t\,(u\cdot v) + t^2\,(v\cdot v)$. The right side is a quadratic in $t$ that never dips below zero, so it cannot have two distinct real roots, so its discriminant obeys $4(u\cdot v)^2 – 4(u\cdot u)(v\cdot v) \le 0$, which is the inequality. Nothing about arrows was used, only the rules of the inner product, so the result holds in every inner product space, the function spaces of the previous footnote included. The definition of angle rests on a theorem, and now you have seen the theorem.

The inner product also settles a practical question we have been finessing: given a vector and a basis, how do you actually extract the components? Call a basis orthonormal if its vectors are mutually perpendicular and of unit length, $e_x\cdot e_x = e_y\cdot e_y = 1$ and $e_x\cdot e_y = 0$. Then dot the expansion $v = v_x\,e_x + v_y\,e_y$ with $e_x$ and let the orthonormality do the sorting:

$$ e_x\cdot v = v_x\,(e_x\cdot e_x) + v_y\,(e_x\cdot e_y) = v_x. $$

A component is a dot product with the corresponding basis vector, $v_x = e_x\cdot v$ and $v_y = e_y\cdot v$: to find how much of $v$ lies along a reference direction, project onto it. And this closes the loop left open with the violin note. Extracting a Fourier coefficient is this same projection, executed with the integral inner product in the space of functions,

$$ c_n = \langle f, e_n \rangle, $$

one coefficient per basis function, one integral each. The formula engineers use daily to decompose a signal is $v_x = e_x\cdot v$ wearing an integral sign.

A vector space with no inner product installed, meanwhile, is still a complete, functioning vector space; its members add, scale, form bases, transform. They simply have no lengths, because no one has defined any. Magnitude and direction, the two pillars of the high-school definition, turn out to be features of one optional attachment.

So, standing where my professor stood, here is how I would now answer his question, in layers. At the foundation, a vector is a member of a vector space: any object, arrow or polynomial or sound wave or quantum state, that can be added to its fellows and scaled by numbers under the usual rules. In physics we ask for one thing more: that the numbers describing the object in different frames be linked by the transformation law, because the object exists and our axes do not. And at the top, as a special case, sits the beloved picture, the arrow in two or three dimensions with an inner product installed, where magnitude and direction happen to capture everything. Running underneath all three layers is the mechanism this article has really been about: change the basis and the components change in the opposite way, so that their combination, the object itself, does not move. If one sentence is worth carrying out of the room, it is that one. The high-school definition is not wrong. It is a portrait of one member of the family, mistaken for the family itself.

I think that is what the smile meant. He knew the definition I gave him was the beginning of the story, and he knew the story was the same one mathematics always tells. The naturals ran out and gave us $-2$; the rationals ran out and gave us $\sqrt 2$; the reals ran out and gave us $i$; and one number per quantity ran out the moment somebody asked which way, and gave us this. Nor does the story stop here. There are quantities in physics that outgrow vectors in turn: the stress inside a loaded beam answers every direction you probe it with by handing back a force vector, and the machine doing the handing back is no longer a vector but a tensor. The metric $g_{\mu\nu}$ from the previous post is exactly such a machine.

The language runs out again. It always does. Vectors are enough to describe quantities with one directional character, but physics soon confronts objects that relate several directions at once. Stress is one example; the metric $$g_{\mu\nu}$$ from the previous posts are another. And those objects are tensors. And we have already met them.¹⁵Read more in-depth interpretation of tensors here: https://aronnomirdha.com/expository/990 What vectors reveal is principle underlying the entire ladder: components depend on the language we choose, while the object itself does not.

References and Footnotes

• 1
  The ancient Greek discovery that the diagonal of a unit square cannot be expressed as a ratio of two integers was one of the earliest demonstrations that the rational numbers are incomplete.
• 2
  The parallelogram rule for composing forces was stated clearly by Simon Stevin in 1586 and appears as a corollary in Newton’s Principia, so the geometric content of vector addition is centuries older than the word. The word itself comes from William Rowan Hamilton in the 1840s, from the Latin vehere, to carry, as part of his quaternions; the modern, stripped-down vector analysis was carved out of the quaternion system by Gibbs and Heaviside toward the end of the nineteenth century. It is a pleasing echo of the professor’s story: Hamilton was searching for a three-dimensional analogue of the complex numbers, which had themselves been invented when the real numbers ran out.
• 3
  The two identities used here are $\cos(\varphi-\theta) = \cos\varphi\cos\theta + \sin\varphi\sin\theta$ and $\sin(\varphi-\theta) = \sin\varphi\cos\theta – \cos\varphi\sin\theta$. They are the whole trigonometric input of this article; everything else is substitution.
• 4
  A bookkeeping note: throughout, the walk is held fixed and the axes rotate, which is called a passive transformation. One can instead hold the axes fixed and rotate the object by $\theta$, an active transformation; the formulas are the same with $\theta$ replaced by $-\theta$. Both conventions are common, which is why signs in other books may differ from mine.
• 5
  I am borrowing a fact here: that a displacement in the plane determines its east part and north part uniquely. Geometrically this is clear enough. In the general setting it is exactly the uniqueness of representation in a basis, which I prove properly once bases are defined below, at which point this step stops being a loan.
• 6
  This opposition has standard names: components that transform inversely to the basis are called contravariant, and objects that transform along with the basis, as the gradient does, are called covariant. The raised and lowered indices of the previous post are precisely this bookkeeping, $x^\mu$ upstairs for contravariant components and $\partial_\mu$ downstairs for covariant objects, and the convention exists so that a paired upper and lower index always assembles a language-independent quantity.
• 7
  For rotations, $R$ is an orthogonal matrix: its rows are the new axis directions written in the old language, it satisfies $\sum_i R_{ij}R_{ik} = \delta_{jk}$, where $\delta_{jk}$ equals $1$ when $j=k$ and $0$ otherwise. This orthogonality is forced rather than assumed: demand that $\sum_i v_i’^2 = \sum_j v_j^2$ for every vector, substitute the transformation law to get $\sum_{j,k}\left(\sum_i R_{ij}R_{ik}\right)v_j v_k$, and the only way the result can equal $\sum_j v_j^2$ for all choices of the $v_j$ is for the bracketed sum to be $\delta_{jk}$. The two-dimensional cancellation we did by hand is the $n=2$ case. Requiring in addition that the frame is not mirror-reflected fixes $\det R = +1$.
• 8
  In modern language: rotations compose to give rotations, every rotation has an inverse, and doing nothing counts as a rotation, so the collection of all rotations forms a group, written $SO(2)$ in the plane and $SO(n)$ in $n$ dimensions. “Transforms like a vector” is, said fully, a statement about how an object responds to this entire group at once, and group theory is the systematic study of such consistency requirements. It becomes one of the load-bearing structures of modern physics.
• 9
  The failure is a finite-angle effect. Rotations through very small angles do commute, up to corrections of second order in the angles, and this is why angular velocity, which is built from infinitesimal rotations, is a legitimate vector while “the rotation itself” is not. There is a further subtlety I am deliberately postponing: quantities like angular velocity behave correctly under rotations but pick up an extra sign under mirror reflection, which is why they are called axial vectors or pseudovectors. That distinction deserves its own discussion.
• 10
  The rules, stated compactly: addition is commutative and associative, there is a zero element, every element has a negative, scaling distributes over both vector addition and number addition, scaling is compatible with multiplication of numbers, and scaling by $1$ does nothing. Eight axioms in all. The “numbers” doing the scaling can be the real numbers or the complex numbers; quantum mechanics is built on complex vector spaces, and everything in this article goes through in either case.
• 11
  Independence here means: if $c_0 + c_1 x + \cdots + c_n x^n$ is the zero vector of this space, which is the polynomial equal to zero for every value of $x$, then all the $c_i$ vanish. Set $x=0$ and $c_0$ falls. Differentiate once, set $x=0$ again, and $c_1$ falls. Keep differentiating; each coefficient drops in turn. So the monomials carry no redundancy, and the components below are the unique ones.
• 12
  A basis is far from unique, but a theorem guarantees that every basis of a given space contains the same number of elements, and that shared number is the dimension of the space. Displacements in a plane have dimension two, displacements in the room have dimension three, and the polynomials and the wave-equation solutions above have infinite dimension, which changes some technical details and none of the conceptual ones.
• 13
  To make “component along a basis function” precise you need an inner product on the function space, and the natural one is an integral, $\langle f, g\rangle = \int f(x)\,g(x)\,dx$ over the interval. The sines and cosines are orthogonal with respect to it, which is what makes the coefficients easy to extract, one integral each. Infinite dimensions bring genuine subtleties about convergence that I am stepping over here, but the geometric picture, a function as a vector and its Fourier coefficients as components, is exactly right.
• 14
  That last formula only defines an angle if the quantity $u\cdot v/(|u||v|)$ actually lies between $-1$ and $1$, and the guarantee is the Cauchy–Schwarz inequality, $(u\cdot v)^2 \le (u\cdot u)(v\cdot v)$. The proof is short enough to give in full. For any number $t$, the vector $u – t\,v$ has a non-negative dot product with itself, so $0 \le (u – t\,v)\cdot(u – t\,v) = u\cdot u – 2t\,(u\cdot v) + t^2\,(v\cdot v)$. The right side is a quadratic in $t$ that never dips below zero, so it cannot have two distinct real roots, so its discriminant obeys $4(u\cdot v)^2 – 4(u\cdot u)(v\cdot v) \le 0$, which is the inequality. Nothing about arrows was used, only the rules of the inner product, so the result holds in every inner product space, the function spaces of the previous footnote included. The definition of angle rests on a theorem, and now you have seen the theorem.
• 15
  Read more in-depth interpretation of tensors here: https://aronnomirdha.com/expository/990