Quaternion Multiplication

While the numbers we use daily follow familiar rules, what happens when we try to extend concepts like multiplication into higher dimensions? This was the question that obsessed the mathematician William Rowan Hamilton, leading to his discovery of quaternions—a four-dimensional number system. The key to their power, and their strangeness, lies in sacrificing a fundamental law of arithmetic: commutativity, where the order of multiplication matters. This article delves into the elegant world of quaternion multiplication. We will first explore the core principles and mechanics of this non-commutative algebra, uncovering how it ingeniously encodes the fundamental operations of 3D vector geometry. Following this, we will journey through its diverse applications and interdisciplinary connections, discovering how this single mathematical framework provides a master key for describing rotations in fields from computer graphics and quantum mechanics to aerospace engineering.

Now that we have been introduced to the strange and wonderful world of quaternions, let us roll up our sleeves and look under the hood. How does this machine actually work? The rules of the game, like the rules of chess, are simple to state but lead to a universe of profound and beautiful consequences. The heart of the matter, the source of all the magic, lies in how two quaternions multiply.

Let's begin with the absolute bedrock of quaternion algebra, the rules that govern its fundamental units, $\vec{i}$, $\vec{j}$, and $\vec{k}$. The Irish mathematician William Rowan Hamilton, in a flash of genius, carved these rules into the stone of Brougham Bridge: $\vec{i}^2 = \vec{j}^2 = \vec{k}^2 = \vec{i}\vec{j}\vec{k} = -1$

From this single, elegant inscription, everything else flows. If you multiply the full expression $\vec{i}\vec{j}\vec{k} = -1$ on the right by $\vec{k}$, you get $(\vec{i}\vec{j})\vec{k}^2 = -\vec{k}$. Since $\vec{k}^2 = -1$, this becomes $-\vec{i}\vec{j} = -\vec{k}$, which tells us that $\vec{i}\vec{j} = \vec{k}$. You can play this game all day and uncover the full multiplication table:

$\vec{i}\vec{j} = \vec{k}, \quad \vec{j}\vec{k} = \vec{i}, \quad \vec{k}\vec{i} = \vec{j}$

But wait! What happens if we try to multiply $\vec{j}\vec{i}$? Let’s go back to $\vec{i}\vec{j}\vec{k} = -1$. This time, multiply on the left by $\vec{i}$. We get $\vec{i}^2(\vec{j}\vec{k}) = -\vec{i}$, which simplifies to $-(\vec{j}\vec{k}) = -\vec{i}$, or $\vec{j}\vec{k} = \vec{i}$. That's consistent. But now, what if we try to find $\vec{j}\vec{i}$? Let's take our result $\vec{i}\vec{j}=\vec{k}$ and multiply from the left by $-\vec{i}$. This gives $(-\vec{i})\vec{i}\vec{j} = -\vec{i}\vec{k}$. This simplifies to $-(-1)\vec{j} = -\vec{i}\vec{k}$, so $\vec{j} = -\vec{i}\vec{k}$. Multiplying by $-1$ gives $\vec{i}\vec{k} = -\vec{j}$. Notice something strange? We found $\vec{k}\vec{i} = \vec{j}$ but $\vec{i}\vec{k} = -\vec{j}$.

This is the first great lesson of quaternions: ​​order matters​​. The multiplication is ​​non-commutative​​. Unlike the numbers you've used your whole life, where $5 \times 3$ is always the same as $3 \times 5$, here $ab$ is not generally equal to $ba$. It's like getting dressed: putting on your socks and then your shoes is not the same as putting on your shoes and then your socks.

A useful way to measure this "failure to commute" is with a mathematical object called the ​​commutator​​, defined as $[q_A, q_B] = q_A q_B - q_B q_A$. If the quaternions commuted, this would always be zero. For quaternions, it is often very much not zero. For instance, if we calculate the commutator for two quaternions like $q_A = 2 + 3\vec{j}$ and $q_B = 1 - 4\vec{i} + \vec{k}$, a direct application of the multiplication rules reveals that $[q_A, q_B] = 6\vec{i} + 24\vec{k}$. This non-zero result is a direct, numerical confirmation of the non-commutative nature we've just uncovered.

So, quaternion multiplication is weird. But is it just a mathematical curiosity? Or is there a deeper reason for this structure? Here lies the true beauty. This non-commutative algebra isn't a flaw; it's a feature that perfectly encodes the geometry of our three-dimensional world.

To see this, let's consider a special kind of quaternion called a ​​pure quaternion​​, which has a zero scalar part, like $p = x\vec{i} + y\vec{j} + z\vec{k}$. You might notice this looks an awful lot like a 3D vector $\vec{v} = (x, y, z)$. This is no accident. We can directly map any 3D vector to a pure quaternion.

Now, what happens if we take two such vectors, $\vec{v}$ and $\vec{w}$, represent them as pure quaternions $p_v$ and $p_w$, and multiply them? Let's see: $p_v p_w = (v_1\vec{i} + v_2\vec{j} + v_3\vec{k})(w_1\vec{i} + w_2\vec{j} + w_3\vec{k})$

If we multiply this out, term by term, and apply the fundamental rules ($\vec{i}^2 = -1$, $\vec{i}\vec{j} = \vec{k}$, $\vec{j}\vec{i} = -\vec{k}$, etc.), we find something astonishing. The product $p_v p_w$ is not a pure quaternion. It has a scalar part and a vector part.

The scalar part turns out to be: $\text{Sc}(p_v p_w) = -v_1w_1 - v_2w_2 - v_3w_3$ This is exactly the negative of the ​​dot product​​ of the two vectors, $-\vec{v} \cdot \vec{w}$. The dot product tells us how much the two vectors are aligned. If they are perpendicular, their dot product is zero.

The vector part of the product is: $\text{Vec}(p_v p_w) = (v_2w_3 - v_3w_2)\vec{i} + (v_3w_1 - v_1w_3)\vec{j} + (v_1w_2 - v_2w_1)\vec{k}$ This is precisely the pure quaternion corresponding to the ​​cross product​​ of the two vectors, $\vec{v} \times \vec{w}$! The cross product gives a new vector that is perpendicular to the plane defined by the original two.

So, the full product is a beautiful, compact package: $p_v p_w = -(\vec{v} \cdot \vec{w}) + (\vec{v} \times \vec{w})$ This is a remarkable unification! A single, mysterious algebraic operation—quaternion multiplication—contains within it the two most fundamental operations of 3D vector algebra.

What about the non-commutativity? Let's look at $p_w p_v$. The dot product part doesn't change since $\vec{w} \cdot \vec{v} = \vec{v} \cdot \vec{w}$. However, the cross product is anti-commutative: $\vec{w} \times \vec{v} = -(\vec{v} \times \vec{w})$. So: $p_w p_v = -(\vec{v} \cdot \vec{w}) - (\vec{v} \times \vec{w})$ Now we see the non-commutativity of quaternion multiplication in a new light. It directly reflects the anti-commutativity of the vector cross product. By cleverly combining the products, we can isolate each vector operation. For instance, if we subtract the two equations above, the dot products cancel and we are left with $p_v p_w - p_w p_v = 2(\vec{v} \times \vec{w})$. This gives us an elegant expression for the cross product purely in terms of quaternion algebra: $p_{\vec{v} \times \vec{w}} = \frac{1}{2}(p_v p_w - p_w p_v)$ The commutator we met earlier is not just a measure of abstract non-commutativity; it is the cross product, up to a factor of 2!

Now we can generalize from pure quaternions to any two general quaternions, $q_1 = s_1 + \vec{v}_1$ and $q_2 = s_2 + \vec{v}_2$, where $s_1, s_2$ are the scalar parts and $\vec{v}_1, \vec{v}_2$ are the vector parts. Applying the distributive law and our newfound knowledge of the pure quaternion product gives the full rule: $q_1 q_2 = (s_1 + \vec{v}_1)(s_2 + \vec{v}_2) = s_1 s_2 + s_1 \vec{v}_2 + s_2 \vec{v}_1 + \vec{v}_1 \vec{v}_2$ Substituting our formula for the product of two vector parts, $\vec{v}_1 \vec{v}_2 = -(\vec{v}_1 \cdot \vec{v}_2) + (\vec{v}_1 \times \vec{v}_2)$, we get: $q_1 q_2 = \underbrace{(s_1 s_2 - \vec{v}_1 \cdot \vec{v}_2)}_{\text{New Scalar Part}} + \underbrace{(s_1 \vec{v}_2 + s_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2)}_{\text{New Vector Part}}$ This formula looks a bit intimidating, but it is the engine that drives all of quaternion mechanics. It tells you precisely how to combine any two quaternions.

One of the key properties that makes a number system powerful is the ability to divide. Quaternions form what is called a ​​division algebra​​, which means that for any non-zero quaternion $q$, there exists a multiplicative inverse, $q^{-1}$, such that $q q^{-1} = q^{-1} q = 1$. The inverse of a quaternion $q = s + \vec{v}$ is given by $q^{-1} = \frac{s - \vec{v}}{s^2 + |\vec{v}|^2}$.

This ability to find inverses means we can solve linear equations. Suppose we have the equation $ax=b$ for some unknown quaternion $x$. In the world of real numbers, we'd just divide by $a$. Here, we must be more careful. Since quaternion multiplication isn't commutative, we can't just move things around. To isolate $x$, we must "un-do" the multiplication by $a$ from the left. We do this by left-multiplying both sides by $a^{-1}$: $a^{-1}(ax) = a^{-1}b \implies (a^{-1}a)x = a^{-1}b \implies 1x = a^{-1}b \implies x = a^{-1}b$ The solution is $x = a^{-1}b$. Notice that $ba^{-1}$ would, in general, be a completely different answer!

This principle is crystal clear when we consider a more general function like $f(q) = aqb$ for some fixed, invertible quaternions $a$ and $b$. To find the inverse function, which takes an output $p = aqb$ and gives back the original input $q$, we must peel off $a$ and $b$ in the correct order. We left-multiply by $a^{-1}$ and right-multiply by $b^{-1}$ to get $a^{-1}pb^{-1} = a^{-1}(aqb)b^{-1} = (a^{-1}a)q(bb^{-1}) = q$. Thus, the inverse function is $f^{-1}(p) = a^{-1}pb^{-1}$. This demonstrates beautifully how the non-commutative structure dictates the rules of algebraic manipulation.

We've established that commutativity is not the norm in the land of quaternions. But does anything commute? Is there a special subset of quaternions that commutes with all other quaternions? This set is called the ​​center​​ of the algebra.

Let's hunt for it. Suppose a quaternion $q = a + b\vec{i} + c\vec{j} + d\vec{k}$ is in the center. It must satisfy $qx = xq$ for every quaternion $x$. It's enough to check if it commutes with the basis elements $\vec{i}$, $\vec{j}$, and $\vec{k}$.

If we enforce the condition $q\vec{i} = \vec{i}q$, a little algebra shows that this can only be true if $c=0$ and $d=0$. So, our candidate must be of the form $q = a + b\vec{i}$. Next, enforcing $q\vec{j} = \vec{j}q$ on this simplified form leads to the condition that $b=0$.

The only thing left is $q = a$. A quaternion with only a real part. Does a real number commute with any quaternion? Yes, because scalar multiplication is commutative. So, the center of the quaternions, the set of elements that behave "normally," is just the set of real numbers themselves. The moment you step off the real number line into the higher dimensions of $\vec{i}$, $\vec{j}$, and $\vec{k}$, you enter a world where order is paramount.

Finally, we can ask if there's an even more fundamental way to think about where quaternions come from. It turns out there is, through a beautiful process of construction. We know that a complex number $z = x + iy$ can be thought of as an ordered pair of real numbers $(x, y)$.

In a similar spirit, a quaternion can be constructed as an ordered pair of complex numbers. Let $q = (a, b)$ where $a$ and $b$ are complex numbers. We can define addition component-wise, as you'd expect. But the multiplication rule is what's special. For two such pairs, $p = (a, b)$ and $q = (c, d)$, the product is defined as: $p \cdot q = (ac - d^*b, ad + bc^*)$ where $z^*$ is the complex conjugate. This rule, known as the Cayley-Dickson construction, seems strange at first. But if you work through it, you find it perfectly reproduces the quaternion algebra we've been exploring. For instance, if you let $\vec{i} = (i, 0)$ and $\vec{j} = (0, 1)$, you can check that $\vec{i}^2 = (-1, 0)$, $\vec{j}^2 = (-1, 0)$, and $\vec{i}\vec{j} = (0, i)$, which we can identify with $\vec{k}$.

This construction reveals a grand hierarchy. We start with the reals $\mathbb{R}$. We build the complex numbers $\mathbb{C}$ by taking pairs of reals, and in doing so, we gain the ability to solve equations like $x^2 = -1$, but we lose the property of ordering (you can't say $i > 0$). Then, we build the quaternions $\mathbb{H}$ by taking pairs of complex numbers. We gain the rich geometry of 3D rotations, but as we have seen, we must sacrifice the commutativity of multiplication. Each step up in this ladder of number systems brings new power at the cost of a familiar property. And it is this very loss of commutativity that makes quaternions not just a mathematical curiosity, but the perfect, elegant language for describing the physics of rotation in the world we inhabit.

Beyond the algebraic rules, the practical utility of quaternion multiplication is extensive. It is not an abstract curiosity but a foundational tool that provides insights across a wide range of scientific and technological disciplines. This strange multiplication, born from Hamilton's decade-long quest, turns out to be a master key for describing rotations and unifying geometric concepts.

This section explores these applications. First, it covers the use of quaternions to describe rotations in three-dimensional space, a technique central to computer graphics, molecular dynamics, and aerospace engineering. It then examines how this representation reveals a deep topological truth about the nature of rotation, famously demonstrated by the "belt trick." Finally, it shows how this single algebraic system unifies concepts in vector geometry—like the dot and cross product—and offers a glimpse into the geometry of four dimensions.

Imagine you are a programmer for a video game, an engineer designing a robotic arm, or a chemist simulating a complex molecule. A central task you face is describing an object's orientation in space and how it changes. A common approach is to use three "Euler angles," but this method is cursed with a frustrating problem known as "gimbal lock," where two of your rotational axes can align, causing you to lose a degree of freedom. It’s like trying to steer a ship when the rudder suddenly gets stuck. Another way is to use $3 \times 3$ rotation matrices. This works, but it's computationally expensive; composing two rotations requires multiplying nine numbers by nine numbers and performing many additions.

This is where quaternions come to the rescue. A rotation by an angle $\theta$ about a unit axis $\hat{n}$ can be perfectly captured by a single unit quaternion. And the true magic happens when we compose rotations: if quaternion $q_1$ represents rotation $R_1$, and $q_2$ represents rotation $R_2$, the composite rotation—applying $R_1$ first, then $R_2$—is described by the quaternion product $q_2 q_1$. Instead of a hefty matrix multiplication, we have a clean, compact quaternion multiplication.

This is not just a theoretical convenience; it is the workhorse of modern science and technology. In molecular dynamics, chemists model the tumbling and vibrations of molecules. Using quaternions to describe the orientation of each molecule is vastly more efficient and stable than using matrices. Over millions of integration steps, numerical errors can accumulate. If one uses matrices, they can slowly stop being perfectly orthogonal, leading to simulated molecules that stretch and distort unnaturally. But with quaternions, the only thing that can go wrong is that their norm might drift slightly from 1. The fix is laughably simple: just divide the quaternion's four components by its current magnitude. This one cheap "renormalization" step automatically guarantees the rotation is perfect, avoiding the costly and complex procedures needed to fix a drifting matrix.

The same mathematics that governs a tumbling molecule also governs the bizarre world of quantum mechanics. The "spin" of an electron, a fundamental quantum property, is described not by a simple vector, but by a state that transforms under rotation using exactly this quaternion-like algebra. A sequence of operations on a quantum bit, or "qubit," in a quantum computer is mathematically equivalent to a sequence of rotations. Determining the final state after a series of quantum gates can be achieved by simply multiplying their corresponding quaternions, elegantly revealing the net effect of the computation. From the microscopic realm of quantum spin to the macroscopic world of rigid-body simulation, quaternion multiplication provides a single, unified, and powerful language of rotation.

The connection between quaternions and rotations holds an even deeper secret, one that touches upon the very topological structure of the space of orientations. As we've seen, a rotation is represented by a unit quaternion $q$. But notice that since $(-q)(-q) = q^2$, the quaternion $-q$ generates the exact same rotation as $q$. Every rotation in our familiar 3D space corresponds to two distinct points in the 4D space of unit quaternions. This "two-to-one" relationship has a stunning and physically verifiable consequence.

Imagine a space probe executing a series of maneuvers that eventually returns it to its starting orientation. In the language of geometry, it has completed a "loop" in the space of rotations, $SO(3)$. We can trace this journey in the world of quaternions, starting from the identity quaternion, $q=1$. When the probe completes its loop and returns to the identity rotation, where does the corresponding quaternion path end? Because both $1$ and $-1$ represent the identity rotation, the path can end at either of these points!

This reveals a profound fact: there are two fundamentally different types of loops of rotations. Those whose quaternion path returns to $1$ are "untwisted," while those that end at $-1$ are "twisted." This is the mathematics behind the famous "plate trick" or "belt trick." Hold a plate flat in your hand and rotate it a full $360^{\circ}$. The plate is back to its original orientation, but your arm is twisted. You've completed a loop in rotation space, but it's a "twisted" loop that corresponds to the quaternion $-1$. You cannot untwist your arm without further rotation. Now, rotate the plate another full $360^{\circ}$ in the same direction. Magically, your arm untwists itself and returns to its original state. This full $720^{\circ}$ rotation is an "untwisted" loop, and its quaternion path has finally returned to $1$.

A complex sequence of rotations, like a probe maneuvering near an asteroid, might seem to bring it back to where it started, but quaternion multiplication can reveal if it's in a topologically "twisted" state relative to its environment. By multiplying the quaternions for each distinct maneuver, we can calculate the final quaternion. If the product is $-1$, the probe has executed a path akin to the $360^{\circ}$ rotation of the plate; it is oriented the same, but its relationship to its history is fundamentally different. The simple act of quaternion multiplication allows us to probe the weird and wonderful topology of the world we live in.

Hamilton's original motivation was to find a way to multiply and divide three-dimensional vectors. What he found was something much richer. When we take two pure quaternions (those with a zero scalar part, which we can identify with vectors in $\mathbb{R}^3$), say $p_1 = \vec{v}_1$ and $p_2 = \vec{v}_2$, their product is:

This is remarkable! A single operation, quaternion multiplication, elegantly packages two fundamental vector operations. Its scalar part gives us the negative of the dot product, and its vector part gives us the cross product. This is not a coincidence; it is a sign of a deeper unity.

This unifying power extends further. Consider three vectors forming the edges of a parallelepiped. Its signed volume is famously given by the scalar triple product, $\vec{v}_1 \cdot (\vec{v}_2 \times \vec{v}_3)$. In the language of quaternions, this geometric quantity is revealed with astonishing simplicity: it is just the negative of the scalar part of the triple product of the corresponding pure quaternions, $-\text{Sc}(p_1 p_2 p_3)$. What were once separate rules in vector algebra are now revealed as different facets of a single, more fundamental algebraic structure.

This elegance provides novel ways to solve geometric problems. For instance, how can we check if three points $P_1, P_2, P_3$ lie on a straight line? In quaternion algebra, we can represent the vectors between them as pure quaternions, say $u = p_2 - p_1$ and $w = p_3 - p_1$. The points are collinear if and only if the quaternion product $(p_2 - p_1)(p_3 - p_1)^{-1}$ is a simple real number. A geometric property is translated into a purely algebraic one, often simplifying the problem immensely.

Perhaps the most mind-expanding application of quaternion multiplication is that it gives us a window into four-dimensional space. A quaternion $q = q_0 + q_1 i + q_2 j + q_3 k$ is naturally an object with four components. The set of all quaternions forms a 4D vector space, $\mathbb{H}$.

Now, let's consider what multiplication does in this 4D space. Take any two quaternions $p$ and $q$, and multiply them by a fixed unit quaternion $u$. Let's say we form $p' = pu$ and $q' = qu$. A remarkable thing has happened: the length of $p'$ is the same as the length of $p$, the length of $q'$ is the same as the length of $q$, and most surprisingly, the 4D "angle" between $p'$ and $q'$ is identical to the angle between $p$ and $q$. In the language of geometry, right-multiplication by a unit quaternion is an orthogonal transformation in 4D space—it is a 4D rotation!.

This can be made even more concrete. The action of multiplying any quaternion $x$ on the left by a fixed quaternion $q$ (the map $x \mapsto qx$) is a linear transformation on the 4D space of quaternions. As such, it can be represented by a $4 \times 4$ matrix. When $q$ is a unit quaternion, this matrix is an orthogonal matrix, an element of $O(4)$. So, the very algebra we use to describe 3D rotations is, from a higher vantage point, a way of performing rotations in 4D.

This 4D nature is intrinsic to the structure of quaternions. Consider the complex numbers, $\mathbb{C}$, which we can view as the subset of quaternions of the form $a+bi$. If we try to multiply a complex number by a general quaternion, say by $j$, the result is typically not another complex number. For instance, $j \cdot i = -k$. The multiplication operation "kicks us out" of the 2D complex plane and into the full 4D space of quaternions.

So, we have come full circle. We began with Hamilton's search for a 3D analogue to the complex numbers and found an algebra that was inextricably 4-dimensional. We then discovered that this 4D system was the perfect tool for describing rotations back in our 3D world. It is a stunning testament to the interconnectedness of mathematics that the rules $i^2 = j^2 = k^2 = ijk = -1$ can describe the spin of an electron, the tumble of a molecule, the path of a spacecraft, the topology of a twisted belt, and the rotations of a 4-dimensional universe, all at once.