Why is the Cross Product Used in Mathematics? Understanding its Role and History

[Trying2Learn]

TL;DR Summary

The question says it all.

So I do know that there does exist a generalization of the cross product (the exterior product), but this question does not concern that (and I would prefer it not )

I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only. (Why does he say this? Simply because it fails associativity?)

I am aware that the operation has two vectors as input, takes their magnitude and the angle between them and outputs a vector that is perpendicular to both and contains information about the angle and the magnitudes.

I can REASON out why this strange operation is so useful when constructing the "moment." I can reason out why it can deliver information about the "tendency" to keep rotating (angular momentum) and can work the operations to show that the rate of change of angular momentum is the moment.

All that is fine.

However, despite that, this operation unnerves me, and I do not know why.

I can see (in my mind) how integrals "sum up effects." I can see in my mind, the role of differentiating. I can see the role of the dot product (in 3D space, not its generalizations). However, this operation called the "cross product" seems to be like rabbit pulled out of a hat: "Oh! It's useful! So, let's use it."

Can anyone discuss the role/need/history of this operation. It seems laden with baggage (i.e.: the "right hand rule" to determine the direction of the resulting vector)

Why (how?) did this operation come about? It just seems whimsical (esp. the sine of the angle between the vectors and, say, not the cosine)

Anything?

 

[Dale]

TL;DR Summary: The question says it all.

"Oh! It's useful! So, let's use it."

Is any other justification needed?

 

[Trying2Learn]

Is any other justification needed?

OK, that was good. That was funny. Put a smile on my face.

But could you elaborate?

In my senses, I see rotations and tendencies to rotate.

In my mind, I see a paraphernalia of mathematical tools.

Fine, then along comes this strange operation: norm of the first vector, norm of the second, the SINE of the angle, make the final vector perpendicular to both, and then (icing on the cake of confusion), use the right hand rule.

This is so infused with an ostensible whimsy, that it baffles me on how this came about.

Or should I just stop thinking and just use it?

 

[Dale]

But could you elaborate?

We use it because it is useful. Its weirdness is not relevant. Using it, regardless of its weirdness, allows us to accurately predict the outcome of experiments and build useful devices like communication systems. That is all that is needed.

 

[Trying2Learn]

We use it because it is useful. Its weirdness is not relevant. Using it, regardless of its weirdness, allows us to accurately predict the outcome of experiments and build useful devices like communication systems. That is all that is needed.

OK, be that way ;-) (joking here)

How did they stumble upon this operation?

Did they just try everything under the sun? Why not the cosine of the angle between the vectors.

So, yes, it is useful. But how did they know in advance (when constructing this operation) that it would be useful?

 

[Ibix]

Why not the cosine of the angle between the vectors.

An operation that produces a vector perpendicular to the plane defined by two vectors cannot work on two parallel or antiparallel vectors. Nor can it be defined if one or other vector has zero magnitude. $|\vec a||\vec b|\sin\theta$ has to be in the running just on that basis, where a cosine-based one is not.

 

[Dale]

But how did they know in advance (when constructing this operation) that it would be useful?

I don’t know the history here, so I cannot comment on that. It is entirely possible that they did not know in advance that it would be useful. Sometimes new math is developed because there is an immediate practical need, like calculus. Sometimes it is developed just because a mathematician liked it, and only later finds a practical use. Sometimes it is useful for one thing immediately, and later it is found to be useful elsewhere. I don’t know how specifically that history played out with the cross product.

Does it matter?

Why not the cosine of the angle between the vectors

I do have to say that I find this part of your reaction a little strange. You are repeatedly harping on the sine rather than the cosine. I don’t get that at all. What is weirder about sine than cosine? Why do you object to the sine? To me that seems the most natural part. The angle is important and it has to go to zero at 0 degrees and be finite at 90 degrees. What else would it be?

 

[vanhees71]

TL;DR Summary: The question says it all.

So I do know that there does exist a generalization of the cross product (the exterior product), but this question does not concern that (and I would prefer it not )

I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only. (Why does he say this? Simply because it fails associativity?)

I've no clue, why he says this. It's nonsense.

The vector product in $\mathbb{E}^3$ has two quite intuitive geometrical meanings.

(a) it measures oriented areas

If you have two linearly independent vectors $\vec{a}$ and $\vec{b}$ by definition $\vec{a} \times \vec{b}$ is a vector perpendicular to the plane spanned by the two vectors (with the direction chosen according to the right-hand rule) with the magnitude given by the area of the parallelogram spanned by the two vectors.

It turns out that the operation is linear in both arguments and that it's antisymmetric, i.e., $\vec{b} \times \vec{a}=-\vec{a} \times \vec{b}$.

(b) it describes "infinitesimal rotations"

If you have a rotation matrix $\hat{D}_{\vec{n}}(\alpha)$, which describes a rotation by an angle $\alpha$ along an axis defined by the unit vector $\vec{n}$ (with the sense of the rotation according to the right-hand rule), then for small $\alpha$
$$\hat{D}_{\vec{n}}(\alpha) \vec{V}=\vec{V} + \alpha \vec{n} \times \vec{V} + \mathcal{O}(\alpha^2).$$

I am aware that the operation has two vectors as input, takes their magnitude and the angle between them and outputs a vector that is perpendicular to both and contains information about the angle and the magnitudes.

The magnitude is, according to meaning (a)
$$|\vec{a} \times \vec{b}|=|\vec{a}| |\vec{b}| \sin \angle(\vec{a},\vec{b}),$$
where $\angle \vec{a},\vec{b} \in [0,\pi]$.

I can REASON out why this strange operation is so useful when constructing the "moment." I can reason out why it can deliver information about the "tendency" to keep rotating (angular momentum) and can work the operations to show that the rate of change of angular momentum is the moment.

That's of course directly related to meaning (b) and is thus imporant to define an angular velocity as an axial vector, $\vec{\omega}$.

All that is fine.

However, despite that, this operation unnerves me, and I do not know why.

Maybe due to the nonsensical statement of Theodore Frankel ;-)). Just use another textbook.

I can see (in my mind) how integrals "sum up effects." I can see in my mind, the role of differentiating. I can see the role of the dot product (in 3D space, not its generalizations). However, this operation called the "cross product" seems to be like rabbit pulled out of a hat: "Oh! It's useful! So, let's use it."

That's true for all the standard operations defined with vectors.

Can anyone discuss the role/need/history of this operation. It seems laden with baggage (i.e.: the "right hand rule" to determine the direction of the resulting vector)

The history is funny. In the beginning, when physicists dealt with (vector) fields they used quaternions, discovered in the beginning of the nineteenth century by Hamilton. Also Maxwell famously formulated his electroamagnetic theory first in terms of quaternions, and it were Heaviside and Gibbs in the beginning of the 19th century who introduced the much more simple notion of vectors in 2D and 3D Euclidean space (which itself is an affine space). If you want to appreciate, how much simpler physics gets with using vectors instead of quaternions, try to read Maxwell's treatise ;-).

Why (how?) did this operation come about? It just seems whimsical (esp. the sine of the angle between the vectors and, say, not the cosine)

The cosine occurs in the dot product, which describes (if one of the vectors is a unit vector) the projection of another vector to the direction of the unit vector, and that's why there's the cosine.

Of course you can also combine the cross and the dot product in the "scalar triple product" (I like the German name "Spatprodukt" better, because of it's intuitive meaning), $(\vec{a} \times \vec{b}) \cdot \vec{c}$. For three non-complanar vectors this gives the volume of the parallelepiped (or "Spat" like a calcite crystal).

Anything?

I hope that helped.

 

[kuruman]

In the article Convenient Equations for Projectile Motion, Am. J. Phys. 29, 623 (1961); doi: 10.1119/1.1937861, J.R. Winans using quaternions (no less) derives the expression $$\mathbf{a} \times\mathbf{s} = \mathbf{v}\times\mathbf{u}.$$ It is applicable to an object moving under constant acceleration $\mathbf{a}$. Here, $\mathbf{u}$ is the initial velocity. The other two quantities, $\mathbf{v}$ and $\mathbf{s}$ are, respectively, the velocity and the position at a later time.

From this, one can derive the horizontal displacement of a projectile $\Delta x$ $$\Delta x=\left|\frac{\mathbf{v}\times\mathbf{u}}{g}\right|.$$ This expression is especially useful not only for finding the horizontal range, but also for maximizing it. The optimization condition for the horizontal range $\mathbf{v}\cdot\mathbf{u}=0$ is obvious.

It is true that everything related to projectile motion can be said starting with the kinematics equations describing the horizontal and vertical component of the position vector as a functions of time. However, doing so will be long and tedious for finding, for example, the projection angle that maximizes the horizontal range of projectile fired from the edge of a cliff of height $H.$

So, yes, it is useful. But how did they know in advance (when constructing this operation) that it would be useful?

I don't know about "them" but I can tell you from my own experience how I happened to come upon the range equation on my own before reading Winans's article. I started by considering the development of an alternate method for solving projectile motion problems using geometry and trigonometry instead of quadratic equations. That work is described here. I used what I call 'the velocity triangle" consisting of the initial and final velocity vectors. Then I considered how one can use the triangle to find what one usually finds in projectile problems, the horizontal range included.

I did not know in advance that the range can be written as the magnitude of a cross product divided by

$g$ but I knew that half of the magnitude of the cross product is the area of the triangle having as sides the two vectors. Looking at the velocity triangle (see figure on the right), it became obvious to me that the range $R=v_{0x}~t_{\!f}$ where $t_{\!f}=\dfrac{v_{fy}-v_{0y}}{g}$ is the area is the area of the triangle divided by $g$. Putting the two thoughts together gave me the cross product equation for the horizontal range. Just to make sure, I verified that $|\mathbf{v}_f\times\mathbf{v}_y|=|v_{0x}(v_{fy}-v_{0y})|.$

To make a long story short, I didn't start out intending to find use for the cross product. I stumbled onto it by diddling with the velocity triangle in order to find an alternate description for projectile motion. As we say in the profession, if we knew what we are doing, it wouldn't be called research.

 

[Filip Larsen]

As an engineer I would like to add, that most of the properties of the vector cross product can at some level be understood by looking at the properties of the equivalent skew-symmetric cross product matrix, $\left[\mathbf a\right]_\times$, such that for two vectors $\mathbf a$ and $\mathbf b$ we have
$$\left[\mathbf a\right]_\times \mathbf b = \mathbf a \times \mathbf b.$$
As an example of how useful this notation can be consider that the rotation matrix for angle $\theta$ around the unit axis $\mathbf n$ can be written as
$$\mathbf R_{\theta\mathbf n} = e^{\left[\theta\mathbf n\right]_\times} = \sum_{i=0}^\infty\frac{\left[\theta\mathbf n\right]_\times^i}{i!} = \mathbf I + sin\theta \left[\mathbf n\right]_\times + (1-cos\theta) \left[\mathbf n\right]_\times^2.$$
That is, the repeated multiplication of a cross product matrix is a cyclic structure such that $\left[\mathbf n\right]_\times$, $\left[\mathbf n\right]_\times^2$, $\left[\mathbf n\right]_\times^3 = -\left[\mathbf n\right]_\times$, and $\left[\mathbf n\right]_\times^4 = -\left[\mathbf n\right]_\times^2$, with $\mathbf n$ being a unit vector. By the way note that $$-\left[\mathbf n\right]_\times^2 = \mathbf I - \mathbf n \mathbf n^T$$ is the plane projection matrix with $\mathbf n$ being the plane normal.

I have earlier considered the vector cross product as a kind of "bastard" operator that quickly could mess up an otherwise beautiful equation, but using the cross product matrix instead I found some of the elegance in notation can been regained, and overall I found the the cross product matrix much more useful (for the engineering work I do, your milage may of course vary).

 

[Kontilera]

To answer very roughly and with hand waving, this is how I see it, in the big picture.

Given two vectors you can either be interested in properties that these vectors has relative to each other. With this I for example mean such aspect as projections onto the other vectors direction. Or, you can be interested in properties associated with the geometrical object they span, such as area, orientation etc.

For the former kind of properties the scalar product is usually involved, whereas for the ladder the cross product is usually involved.

Now, two vectors span a parallellogram, in the most obvious of ways, which is a two dimensional object. In R^3 however, neglecting the shape, every oriented parallellogram can be describe by a vector perpendicular to it. The direction of the vector describes the orientation and angles of the parallellogram and the length represents the area.

This is however only in R^3, since this is the only case where the number of dimensions (x, y and z) coincide with the number of basis planes (xy, yz, zx).

In R^4 you have (x, y, z, t) vs (xy, yz, zt, tx, xz, ty), thus vectors are not suitable to represent two dimensional objects.

PS. Sorry for my english. DS.

 

[DaTario]

In the hope that it will be possible today to translate an article in PDF format from Portuguese to English, I think this reference may be useful. It tells us that the historical origin of the cross product is in Hamilton's work on quaternions.

https://www.scielo.br/j/rbef/a/XVRqzXNzJHmwDKsQbPqHMFw/?format=pdf&lang=pt

But there is one feature of this operation that bothers me too. From the point of view of one of the possible algorithms for carrying out the operation (there may be others, I don't know), it has the structure of a calculation of the determinant of a 3x3 matrix. However, the matrix elements of the second and third rows are coordinates, while the matrix elements of the first row are unit vectors. This strange lack of parallelism in this operation leads me to classify this product as a rather unique operation.

 

[pines-demon]

Also Maxwell famously formulated his electroamagnetic theory first in terms of quaternions, and it were Heaviside and Gibbs in the beginning of the 19th century who introduced the much more simple notion of vectors in 2D and 3D Euclidean space (which itself is an affine space). If you want to appreciate, how much simpler physics gets with using vectors instead of quaternions, try to read Maxwell's treatise ;-).

Maxwell did not use quaternions in his key publications on Maxwell's equations of 1865 and 1873, he just wrote component by component. Did I miss some publication where he uses quaternions?

 

[pasmith]

From the point of view of one of the possible algorithms for carrying out the operation (there may be others, I don't know), it has the structure of a calculation of the determinant of a 3x3 matrix. However, the matrix elements of the second and third rows are coordinates, while the matrix elements of the first row are unit vectors. This strange lack of parallelism in this operation leads me to classify this product as a rather unique operation.

I regard this as an abuse of notation. It is used because it happens to give the correct result, not because it makes mathematical sense.

In the same way, the curl can be computed in cartesian coordinates by means of a determinant in which the first row are unit vectors, the second are differential operators, and the third are functions. This does not give the correct answer in curvilinear coordinates.

 

[renormalize]

This does not give the correct answer in curvilinear coordinates.

Are you saying that these entries from http://hyperphysics.phy-astr.gsu.edu/hbase/curl.html don't work?

 

[Ibix]

They work, but the first rows aren't unit vectors.

 

[mathwonk]

In my copy from Dover, of the 1891 third edition of Maxwell's Electricity and Magnetism, published of course after he had died, and proofed by Thomson, the "preliminary" discussion of mathematical theorems, including what we call Stokes's theorem, uses quaternions. In particular, the operator "del" defined as i∂/∂x + j∂/∂y + k∂/∂z, is manipulated (bottom p.29) by "remembering the rules for multiplication of i,j,k". In particular, on the last page, p.30, the formula del^2 = minus the Laplace operator, depends on the rules i^2 = j^2 = k^2 = -1, ij =k, ji = -k, etc... as does the fact that the usual divergence appears there with a minus sign and called therefore the "convergence".

I do not know how these formulas appear in earlier editions. And what I am calling "del" seems to be "nabla" or "inverted delta".

 

[Mister T]

I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only. (Why does he say this? Simply because it fails associativity?)

Do you have a reference for this?

I can see (in my mind) how integrals "sum up effects." I can see in my mind, the role of differentiating. I can see the role of the dot product (in 3D space, not its generalizations). However, this operation called the "cross product" seems to be like rabbit pulled out of a hat: "Oh! It's useful! So, let's use it."

I don't see how the notion of the cross product being useful is any different than those other things being useful.

Did they just try everything under the sun? Why not the cosine of the angle between the vectors.

So, yes, it is useful. But how did they know in advance (when constructing this operation) that it would be useful?

All of those issues arise with integrals, derivatives, and the dot product. Why do they take the form they take? Because that's the form that's useful. Why use a derivative to describe velocity? Because the derivative describes velocity. Yes, they tried other formulations such as taking the derivative of velocity with respect to distance instead of respect to time to define acceleration. The latter gained more popularity because it's more useful.

These things weren't discovered carved into stone tablets that had been buried. They are inventions of the human mind.

 

[A.T.]

I can see the role of the dot product (in 3D space, not its generalizations).

The roles of dot and cross products in physics are somewhat complimentary:

The dot product is zero when the vectors are perpendicular, and maximal when they are parallel. The cross product (magnitude) has the opposite properties of the above.

So you choose one or the other depending on what observed physical relationship you want to describe.

The cross product also has a direction. But note that in many cross product applications, one of the 3 involved vectors has a direction chosen purely by convention, just to make the cross product work. And conversely, the result of the dot product is often used to scale a unit vector.

 

[mathwonk]

@Trying2Learn: I hope this helps motivate the cross product.

Cross products, Dot products, Determinants

Computations in the plane:
A computational technique is best understood in terms of the problem it is designed to solve. In the plane, basic questions about vectors are to calculate the length of a vector, and the area of the parallelogram spanned by two vectors, both in terms of the coordinates of the vectors. The pythagorean theorem solves the first, and determinants solve the second.

If v = (a,b) is a plane vector, then its squared length |v|^2 equals a^2+b^2, by the Pythagorean theorem.

If w= (x,y) is a second plane vector, then the area of the parallelogram spanned by v, w (base v, and side w oriented counterclockwise from v) equals the determinant ay-bx.

It is also of interest to compute the projection of w onto (the line through) v, as well as the projection of w (onto the line) perpendicular to v. This last projection of course also helps compute the area of the parallelogram spanned by v,w, since that area equals the length of v times the length of this projection (base times height).

In terms of trig functions, the projection of w onto v, has length |w|cos(t), where t is the angle ≤ π between v and w, and the projection of w onto the line perpendicular to v has length |w|sin(t). Thus the parallelogram spanned by v and w has area equal to |v||w|sin(t) = ay-bx. Thus the projection of w perpendicular to v has length (ay-bx)/|v|.

Using the law of cosines, Euclid’s generalization of the Pythagorean theorem, one can compute that |v||w|cos(t) = ax+by = v.w, the dot product of v and w. Thus we can compute the length of the projection of w onto v as (v.w)/|v|.

Computations in three space:
If v = (a,b,c) and w = (x,y,z) are vectors in 3-space, Pythagoras generalizes to give the length of v as |v| = a^2+b^2+c^2, i.e. the squared length of v equals the sum of the squared lengths of the three projections of v onto the three axes.

Now it is quite interesting that the area of the parallelogram [v,w] spanned by v,w shares a similar property: i.e. the squared area of [v,w] equals the sum of the squares of the three projected parallelograms into the three coordinate planes, (y,z), (x,z), and (x,y).

Thus (area[v,w])^2 = (bz-cy)^2 + (az-cx)^2 +(ay-bx)^2.
This, by usual Pythagoras, equals the length of the vector
vxw = (bz-cy, cx-az, ay-bx), i.e. (area([v,w])^2 = |vxw|^2.

By our choice of sign in the second entry, we also have that this vector vxw is perpendicular to both v and w, and as a vector, it equals the determinant of the 3x3 matrix with unit vectors i,j,k in the first row, and the coordinates of v,w in the next two.

Thus the cross product vxw is obviously a beautiful tool, providing solutions to several interesting and basic problems concerning vectors in 3-space, which everyone may celebrate and enjoy.

By the way, since Professor Frankel is highly sophisticated in mathematics, especially differential geometry, and has written a book, The Geometry of Physics, intended to illustrate the superiority of the language of differential forms over classical vector analysis, I suspect, If he made that disparaging remark, it was merely to emphasize that he personally recommends forms as much more useful and flexible a tool.

Recall also that at various times in history, complex numbers have been called imaginary, sophistical, absurd, and impossible. and "irrational" numbers and square roots are called (ab)surds.

 

[Hornbein]

The geometric product of geometric algebra works in any number of Euclidean dimensions and has the same effect as the cross product. The cross product is more concise and physics doesn't care about other numbers of dimensions, so that's why it's used.

 

[mathwonk]

From what I have read, in seeking to understand how the cross product arose, it seems that curiosity, such as yours, played a big role. In the 19th century people were aware that Gauss and Riemann had showed how to define a multiplication on R^2, the complex numbers a+bi, and it had proved useful. So Hamilton wondered if he could define an extension, multiplication on elements of R^3, and what properties it should have. Apparently he first considered generalized complex numbers like a+bi+cj, which would contain the reals. He chose to have i^2 = j^2 = -1, but what to define ij as? He eventually realized he needed it to be a 4th object k, and that, especially if he wanted to have associativity, he needed to define his multiplication on R^4 not R^3, on elements like a + bi+cj+dk, where if ij = k, then iij = -j = ik, ijk = -1, ...and so on.

If we just restrict to elements ai+bj+ck, with again ij=k, jk = i, ki = j,....., then when we multiply these by these(quaternion) rules, we will get also some scalars, and if we want to exclude those scalars, to obtain a multiplication just on R^3, we must set i^2 = j^2 = k^2 = 0. Then we lose associativity, since now i(ik) = i(-j) -k, = 0, but (ii)k = 0. These are the rules for the cross product.

If we return to the full quaternion multiplication, then product of two "pure" quaternions, of form bi+cj+dk, is no longer pure, but has both a scalar and a vector part, namely (minus) the scalar product, and the vector (or cross) product.

So apparently mathematicians just asked themselves what are the possible multiplications on objects like bi+cj+dk, or a+bi+cj+dk, noticing that all they needed to decide was what should be the products of the i,j,k. Only a few reasonable possibilities exist, obeying some usual properties. Then apparently after these multiplications were constructed, they were seen to be useful in expressing physical phenomena, which solidified their role.

Here is the most detailed account of their history I have found:
https://math.stackexchange.com/questions/62318/origin-of-the-dot-and-cross-product