A plane disguised as an arrow

The dot product keeps one thing about a pair of arrows and discards the rest. What it keeps is a shadow: how far one arrow reaches along the other, times the other’s length. What it discards is everything perpendicular to that — how far the first arrow reaches away from the second, and in which direction.

In three dimensions there is a second product that keeps the discarded part. It takes two arrows $a$ and $b$ and returns a third, $a \times b$, perpendicular to both, pointing to the side fixed by the right-hand rule, and exactly as long as the area of the parallelogram the two arrows span.

Everything about that description works, and it has been used for a hundred and forty years to compute torques, normals to surfaces and magnetic forces. It is also, in a precise sense, a disguise. The thing the product really computes is not an arrow at all, and the three dimensions of ordinary space are the only place the disguise fits.

Three numbers that are three shadows #

The formula for the product is usually memorised as a determinant with unit vectors in its first row, which produces

$$a \times b = (a_2b_3 - a_3b_2,\; a_3b_1 - a_1b_3,\; a_1b_2 - a_2b_1),$$

and which explains nothing. Each of the three entries has a meaning, and the meaning is an area.

Take the parallelogram on $a$ and $b$ and project it straight onto the plane of the second and third coordinates — throw away the first coordinate of every point. What is left is a parallelogram in a plane, spanned by $(a_2, a_3)$ and $(b_2, b_3)$, and its signed area is $a_2b_3 - a_3b_2$ — the first entry of the product. The second entry is the shadow on the plane of the third and first coordinates, the third is the shadow on the plane of the first and second.

The signs are not a nuisance. A shadow has an orientation — whether going from $a$ to $b$ turns anticlockwise or clockwise in that plane — and the sign records it. The cross product is a list of three oriented areas, one for each way of looking at the parallelogram along a coordinate axis.

And the three shadows obey a Pythagoras of their own. Their squares add to the square of the parallelogram’s true area: here about $8.23$, the square of $2.869$. That is the two-dimensional version of the familiar fact that a segment’s squared length is the sum of its squared shadows on the axes, and it has a classical special case — de Gua’s theorem, that the squared area of the slanted face of a right-angled corner of a box equals the sum of the squared areas of the other three faces.

The length is what Cauchy–Schwarz left over #

Adding up the squared shadows gives the product’s squared length, and there is a second way to compute the same number that connects it straight back to the dot product.

$$|a \times b|^2 = |a|^2|b|^2 - (a\cdot b)^2 .$$

That is Lagrange’s identity, and it is the exact equation behind Cauchy–Schwarz: the inequality $(a \cdot b)^2 \le |a|^2|b|^2$ holds because the difference between the two sides is a sum of squares, and the squares being summed are the three components of $a \times b$. Written with an angle it says $|a \times b| = |a||b|\sin\theta$, beside $a \cdot b = |a||b|\cos\theta$.

So the two products split one quantity between them. $|a|^2|b|^2$ is shared out: the dot product takes the part along, the cross product takes the part across, and the squares of the two add back to the whole. Two arrows pointing the same way put everything into the first; two at right angles put everything into the second.

That also says why the length is an area rather than a coincidence of notation. Base times height is $|a|$ times the part of $b$ perpendicular to $a$, and the perpendicular part is what is left of $b$ after its shadow on $a$ is removed — the subtraction that clears a direction, which is exactly how the figure measures the area it compares against. The Gram determinant is the same computation written for any number of arrows in any number of dimensions, and it never needs a cross product to do it.

A mirror gives it away #

An arrow that describes a real displacement — the difference between two positions — behaves in a mirror exactly as it looks as though it should: its reflection is the displacement between the reflected positions. Test the cross product the same way and it fails.

For any reflection $M$, $(Ma) \times (Mb) = -M(a \times b)$, and for any rotation the sign is a plus. The general rule is that the product picks up the determinant of the map, which is $+1$ for a rotation and $-1$ for a reflection. An honest arrow would transform with no such factor.

The failure is a direct consequence of the shadows. A reflection reverses orientation — a clockwise turn in the mirror is anticlockwise in the room — so it flips the sign of oriented areas, and the three entries of the product are oriented areas. The product is correct in the mirror world as a plane; it only looks wrong because the rule for turning a plane into an arrow — the right-hand rule — is itself a choice of orientation, and the mirror reverses the choice.

Physics calls such quantities axial vectors, or pseudovectors. Angular momentum, torque and the magnetic field are all of this kind, and for most of the history of physics the distinction was a matter of bookkeeping, because the laws were believed to look the same in a mirror. In 1957 Wu’s experiment with cobalt-60 showed that the weak interaction does not: electrons emerged preferentially on one side of the nuclear spin, an axial vector, and a mirror image of the apparatus would have shown them on the other. That was the first time the difference between an arrow and a plane disguised as one could be read off a laboratory bench.

What the arrow is standing in for #

If the product is really a plane, the plane is the object worth writing down, and it can be written down in any number of dimensions.

The oriented plane spanned by $a$ then $b$ — called their wedge product, $a \wedge b$ — is recorded by its shadows on every coordinate plane: one number $a_ib_j - a_jb_i$ for each pair of coordinates $i < j$. That is a list of oriented areas, it needs no right-hand rule, and it behaves correctly under every map, reflections included.

In three dimensions there are three coordinate planes and therefore three shadows, and each coordinate plane leaves out exactly one axis. Matching each shadow to the axis its plane leaves out turns the three oriented areas into three coordinates of an arrow, and that matching is the cross product. The disguise is the matching: a plane that leaves out the first axis is filed under the first axis, and so on. The matching needs a choice of which way round each plane is, which is the right-hand rule, and it is precisely that choice a mirror reverses.

The wedge product also shows what the dot product and the cross product have in common. Both are halves of a single object, the product that Clifford and Grassmann built, whose symmetric part is the dot product and whose antisymmetric part is the wedge. Hamilton’s quaternions had the same split inside them from the start: multiplying two pure quaternions gives minus their dot product as the real part and their cross product as the rest.

Why three #

Two facts make the matching possible, and both are arithmetic.

A plane needs as many numbers as an arrow only when $n = 3$. The count of coordinate planes is $n(n-1)/2$, and it equals $n$ only at $n = 3$. In two dimensions a plane needs one number — the single oriented area $a_1b_2 - a_2b_1$, which is the determinant — and there is no arrow of one coordinate to disguise it as. In four dimensions a plane needs six numbers and an arrow has four.

And a plane has a single perpendicular direction only when $n = 3$. In $n$ dimensions the directions perpendicular to a plane form a space of dimension $n - 2$. In three that is a line, so “the” direction perpendicular to $a$ and $b$ makes sense. In four it is another plane, and asking for an arrow perpendicular to both $a$ and $b$ has infinitely many answers pointing in genuinely different directions.

So the two coincidences that make the cross product a well-defined arrow happen in the same dimension. That is why the product feels inevitable in physics, which is done in three dimensions, and why anybody who meets it first there is surprised that it has no four-dimensional version.

A normal is carried differently #

The distinction has a practical consequence that every piece of software drawing lit surfaces has to handle.

A surface patch spanned by two edge arrows has a normal, the cross product of the edges. Apply a linear map $M$ to the whole patch — shear it, stretch it — and the edges become $Ma$ and $Mb$. The obvious guess for the new normal is $M$ applied to the old normal. It is wrong.

The correct rule is $(Ma) \times (Mb) = \operatorname{cof}(M)\,(a \times b)$, where the cofactor matrix is $\det(M)$ times the inverse of the transpose. The determinant is the reflection factor from the mirror, and the inverse transpose is the reason the rule is not $M$ itself.

That inverse transpose is not a coincidence either. A normal is what is left when a plane is described by the one direction perpendicular to it, and perpendicularity is a statement about a product: a normal $n$ is the arrow for which $n \cdot v = 0$ for every $v$ in the plane. Moving $M$ across that product is exactly what the adjoint does, and under the ordinary product the adjoint is the transpose. An arrow is carried by $M$; the thing that measures arrows by a product is carried by the inverse of $M$’s transpose, and a normal is the second kind of object wearing the first kind’s clothes. The graphics convention of a separate “normal matrix” beside the model matrix is this sentence, implemented.

Seven, and what it forgets #

There is one other dimension in which a product with both properties exists — perpendicular to its factors, with a length given by Lagrange’s identity — and the price it pays shows exactly what the three-dimensional product was getting for free.

The construction uses the seven lines of the Fano plane. Name seven unit arrows $e_1, \ldots, e_7$ and declare $e_i \times e_{i+1} = e_{i+3}$, with indices wrapped round modulo seven and the usual cyclic and antisymmetric rules. That is the multiplication table of the imaginary octonions, with the real part discarded.

Three different planes give the same product. In three dimensions the product determines the plane it came from: the plane is everything perpendicular to it. In seven dimensions a plane needs twenty-one numbers, the product has seven, and fourteen numbers’ worth of information has been thrown away. The seven-dimensional cross product is a genuine product, and it is not a record of a plane.

It also loses an algebraic law. In three dimensions $a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$, the Jacobi identity, which is what makes the product the algebra of infinitesimal rotations. In seven dimensions the same sum on three unit arrows comes out non-zero, and the product governs no group of rotations in that way.

Why exactly three and seven? A product with the two properties can be glued to the real numbers to make a multiplication on $n + 1$ dimensions in which lengths multiply, and Hurwitz’s theorem allows such multiplications only in dimensions $1$, $2$, $4$ and $8$. Subtracting the real direction leaves $0$, $1$, $3$ and $7$ — and the first two give products that are identically nought.

The triple product, which is honest #

One combination of the cross product is not disguised at all. Taking its dot product with a third arrow gives

$$a \cdot (b \times c),$$

a single number, and it is the signed volume of the box on the three arrows — the determinant with $a$, $b$ and $c$ as its columns, which is the only function that behaves like a volume.

The volume’s sign records whether the three arrows, in order, make a right-handed frame, and the hero figure uses exactly that to confirm its drawing: $a$, $b$ and $a \times b$ come out with a positive triple product. Under a reflection the triple product changes sign, which makes it a pseudoscalar, and for the same reason: a volume with an orientation is still an oriented object.

What makes the triple product honest is that it does not pretend. It is advertised as a number that depends on orientation, and it does. The cross product is advertised as an arrow and is an oriented plane, and the gap between the advertisement and the object is the whole of what goes wrong with it.

Where the product needs care #

It is not associative. $a \times (b \times c)$ and $(a \times b) \times c$ are generally different arrows. The first expands to $b(a\cdot c) - c(a\cdot b)$, which lies in the plane of $b$ and $c$, and the second lies in the plane of $a$ and $b$. Expressions with two cross products need their brackets.

It has no division. Knowing $a$ and $a \times b$ does not determine $b$: adding any multiple of $a$ to $b$ leaves the product unchanged. The plane is determined and the arrow within it is not.

Nearly parallel arrows lose their digits. The product of two arrows at a tiny angle is the difference of nearly equal numbers in every component, so its direction can be dominated by rounding — the same failure as removing a shadow from a nearly parallel vector. A normal computed from a thin sliver of a triangle is unreliable for exactly this reason.

And the handedness is a convention that software does not agree on. Some graphics systems use left-handed coordinates, in which the right-hand rule becomes a left-hand rule and every cross product computed by a library from the other convention points the other way. The arithmetic is identical; the pictures are mirror images.

Hamilton, Grassmann, and a quarrel about notation #

Hamilton found the quaternions in 1843, and the cross product arrived with them rather than on its own: the product of two quaternions with no real part has minus the dot product as its real part and the cross product as the rest. For forty years physicists who wanted either had to take both.

Grassmann’s Ausdehnungslehre of 1844 had the better object — the exterior product, which is the wedge of the fourth section above, defined in every dimension — and was so badly received that he left mathematics for Sanskrit philology, where his work is still cited.

Gibbs in America and Heaviside in Britain, in the 1880s, independently cut the quaternion product in two and kept the halves as the dot and cross products, because that was what Maxwell’s equations needed. The quaternionists, led by Tait, objected in print through the 1890s, and lost. The notation that won was the one that worked in three dimensions and nowhere else, which is a reasonable choice for physics and the source of every surprise this essay describes.

What a drawing of an arrow cannot show #

Every three-dimensional figure here is a projection onto a page, and a projection can itself reverse orientation — whether a drawn frame looks right-handed depends on the camera as much as on the arrows. The figures check handedness in the arithmetic, not in the picture, because the picture cannot be trusted to show it.

The mirror figure shows two opposite arrows and cannot show why one of them is the right one: both are perfectly good arrows, and the claim that the product’s reflection is wrong is a claim about how the product was defined.

And the seven-dimensional product has no picture at all. It is a table of whole numbers, which is honest, and a reader has to take the forgetting of the plane on the strength of three lines of arithmetic rather than on anything seen.

Still open: the plane in every dimension #

The wedge product is where the cross product goes when the dimension changes. Oriented planes, oriented volumes and their higher analogues form the exterior algebra, the curl of vector calculus becomes the exterior derivative, and the theorems of Green, Stokes and Gauss become one theorem about the boundary of an oriented region. The pairing between a plane and its perpendicular direction that made the cross product possible has a general form, the Hodge star, which needs both an orientation and an inner product — which is to say both of the ingredients this essay found hidden inside the right-hand rule.

Quaternions, which carried the product before it was separated, turn out to be the practical way to compose rotations, and the Jacobi identity that fails in seven dimensions is the start of the story of Lie algebras.

A coincidence that looks like a structure #

The habit is about spotting a construction that works because of a numerical accident.

The cross product works because $n(n-1)/2 = n$ has the solution $3$, and because $n - 2 = 1$ has the same one. Nothing in its usual presentation mentions either equation, and a reader who learns it as a formula has no way to know it is special. The first sign is always the same: some property that ought to be automatic — here, behaving like an arrow in a mirror — fails in a way the formula did not predict.

When a construction seems to need a particular dimension, count what it is really recording and see whether the count agrees with the dimension by accident. The answer usually names the more general object, and the more general object is usually the one that survives.