Classical Hamiltonian quaternions

For the history of quaternions see:history of quaternions

For a more general treatment of quaternions see:quaternions

In geometry, quaternions are a mathematical entity invented by William Rowan Hamilton in 1843. This article is written exclusively in Hamilton's original notation, using his original definitions of terms. It uses only Hamilton as its primary source and reliable secondary sources written on or before 1905. Works on quaternions now span three centuries, and this artificial distinction is an agreed upon editorial device, used to organize material on the subject. Since this article describes Hamilton's original treatment of quaternions, using his own notation and terminology, and definition of terms, this treatment is more geometric than the general treatment of quaternions. Mathematically speaking, the quaternions discussed in this article are the same quaternions that are used in almost all modern applications.

Some other writers use different notations, define terms differently and also define a quaternion differently than Hamilton. The question as to whether the mathematical entity defined by Hamilton is the same as the mathematical entities that other writers have called quaternions, over the centuries, has been at times a hotly debated subject; however, comparison with other notations systems is a topic for other articles.

Historical overview

In classical quaternion notation a unit of distance squared was equal to a negative scalar quantity. The Pythagorean theorem, where B = 3i and C = 4j are the sides of a right triangle and A is the hypotenuse would look like:

$A^{2}=B^{2}+C^{2}\,$

$-25=-9-16\,$

To put it in classical quaternion terminology: the square of every vector is a negative scalar. In 1835, years before he discovered quaternions, Hamilton wrote an article entitled "Algebra the Science of Pure Time", which expressed the view that time worked like a real number or scalar.

Hamilton's discovery of quaternions has sometimes been linked a 19th century version of the modern concept of spacetime. In the words of John Baez, Hamilton gave quaternions a "cosmic significance". As time progressed, Hamilton devoted more and more of his efforts to pure mathematics, continuing his research into quaternions from the time of his discovery of them until his death in 1865.

Classical elements of a quaternion

There are a number of different types of quantities used in Hamilton's quaternion calculus, and understanding their definitions is critical to mastering the subject. Among the four most important kinds of geometric quantities are tensors, vectors, scalars and versors. Each of these four distinct types of geometrical entities has different subtypes with special names. A quaternion is made up of these elements, and can be deconstructed into them. Every quaternion, for example, contains a vector, and a scalar, as well as a tensor and a versor.

In addition to these kinds of geometric quantities are the the operatons. The four fundamental operations are addition, subtraction, multiplication and division. In addition, the T,U,S,V and K operators stand for 'take the tensor of', 'take the versor of', 'take the scalar of' 'take the vector of' and 'take the conjugate of', and can operate on each of the four types of quantities.

This section gives Hamilton's classic definitions for these elemental quantities and the next two sections explain various operations.

Vector

See also: the operation called take the vector of

Hamilton's second book, Elements of Quaternions begins with the following line.

A right line AB, considered as having not only length,
but also direction, is said to be a vector.  Its initial point A
is said to be its origin; and its final point B is said to be its
term.  A vector AB is conceived to be (or construct) the
difference of its two extreme points; or, more fully, to be the 
result of the subtraction of its own origin from its own term;
and in conformilty with this concetion, it is also denoted by
the symbol B - A...

The vector of a quaternion written Vq was a well defined mathematical entity in the classical quaternion notation system.

The square of every vector was equal to a negative scalar. of which the positive opposite expresses the square of the length of the vector.

Hence it can be represented in a trinomial form. This pure vector then consists of only three components each of which is another vector. A vector is a three dimensional entity.

$ix+jy+kz\,$

or

$xi+yj+zk\,$

A vector may also be represented by a Greek letter such as α, β, γ

A vector may be obtained from any quaternion by the operation called taking the vector of, and a vector so obtained is an entity of pure dimension containing no scalar part.

When a vector is multiplied by another vector by the operation of geometric multiplication the result is a quaternion, which consists of both a vector and a scaler part.

When a vector is divided by another vector called geometric division result or quotient is also another quaternion.

One approach to defining a vector is to view it as the subtraction of two different coordinates.The subtraction symbol, when applied between two different coordinates results in a new type of mathematical entity called a vector which represents the act of moving from one point to another. For example if one point is the earth, and the other point is the sun, subtracting the position of the sun to the earth results in a vector, which represents going from the earth to the sun. It can be represented as an arrow drawn from the earth to the sun. It has both magnitude and direction.

Hamilton introduced the world to the concept of a vector in the 1840s. In his first lecture article 15, Hamilton introduces the word vector, from the Latin vection, or to move.

Unit vector

A unit vector is a vector with a length or tensor of one. Examples of unit vectors include i,j and k

Tensor

This section is about use of the word tensor in the context of quaternions, for a more general discussion or the term see tensor.

See also: the operation called take the tensor of

In Hamilton's usage, a tensor is defined as a positive numerical quantity, or more properly signless, number.A tensor can be thought of as a positive scalar The "tensor" can be thought of as representing a "stretching factor."

The term tensor was introduced by Sir William Rowan Hamilton in his first book, Lectures on Quaternions, based on his lectures given shortly after his invention of the quaternions:

• it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length ; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even incommensurable tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraical signs of positive and negative ; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.

When the operation represented with the letter T called Take the tensor of is performed the result is a tensor.

Scalar

Main article: Scalar_(mathematics)

Hamilton invented the term scalars for the real numbers, because they span the "scale of progression from positive to negative infinity". The scalar of a quaternion q is its first component, or real part, denoted Sq.

Versor

Main article: versor

Every radial quotient is a versor. As Hamilton put it:

When a quaternion $q=\alpha :\beta \,$ is thus 
radial quotient, or when the lengths of the two 
lines $\alpha$ and $\beta$ are equal,
the effect of this quaternion q when considered as a Factor
in the equation $q\alpha =\beta \,$, the effect 
is simply the turning of the multiplicand 
line $\alpha \,$ in the plane of q
in the plane of q towards the hand determined by the direction
determined by the positive axis Ax.q through 
the angle denoted by $\angle q$ so as to bring the line $\alpha \,$
(or a revolving line which had coincided where with) into a 
new direction: namely, into that of the product 
line $\beta$.  And with reference to this conceived 
operation of turning, we shall now say that 
every radial quotient is a versor.

The versor of a quaternion which can be written as

$\mathbf {U} q$

is another special type of quaternion with useful properties.

The tensor of a versor is always equal to one.

$\mathbf {TU} q=1$

In general a Versor can be associated with a plane, an axis and an angle.

When a versor and a vector which lies in the plane of the versor are multiplied the result is a new vector of the same length but turned by the angle of the versor.

Vector arc

A versor can also in general be represented by a unique great circle arc, called a vector arc. This arc is greater than zero and less than 180 degrees. This is because the shortest distance between any two points of a sphere has a maximum limit of an arc corresponding to 180 degrees.

Since every unit vector can be thought of as a point on a unit sphere, since a versor can be thought of as the quotient of two vectors, it has a representative vector arc connecting these two points, drawn from the divisor or lower part of quotient, to the dividend or upper part of the quotient.

Right versor

When the arc of a versor has the magnitude of a right angle, then it is called a right versor, a right radial or quadrantal versor.

Like all quaternions a versor can be decomposed into the product of its tensor and its versor.

The versor of a versor is the same as the versor:

$\mathbf {UU} q=\mathbf {U} q$

Like other quaternions, a versor consists of the sum of a scalar and a vector.

An important point to remember is that the unit vector of a right versor, written $\mathbf {UVU} q$, can potentially be much longer than a different but related unit vector corresponding an infinitesimal unit tangent drawn along the vector arc defined by the same right versor.

Three dimensional rectangular unit vectors as Hamilton proved can be thought of as infinitesimal segments of arc length. The corresponding unit vector of the versor that represents this arc has a finite length, but they both have the same multiplication table.

It was well known at the time that finite rotations add differently from infinitesimal ones.

A an example of right versor can be and was written in classical texts as

0 + i

A right versor can be used to represent 90 degrees of arc length. It is a completely different kind of number from a the unit vectors used as the basis of a rectangular coordinate system.

The two are different but they have a very interesting relationship that Hamilton discovered.

As an example give the unit vector that defines an infinitesimal unit of arc a length of one foot. Think of it as a line drawn on the surface of the earth. This one foot line on the earth is an example of a Vector-Arc.

All the versors taken together form a unit sphere. For this example the earth could be thought of as the unit sphere corresponding to the one foot line on the ground. The tensor of the sphere and the tensor of the one foot long unit vector are both one. But the big unit vector, the vector of the versor is the diameter of the earth, and the little unit vector is just one foot long. In other words, two unit vectors don't have to be the same unit of length, when they are used in different contexts.

The relationship between the two as classical texts prove is in the arc length. The zero, in this expression 0 + i really means the limit as a right versor approaches zero. Think of two lines drawn from the center of the earth to each of the ends of a one foot line on the ground. This angle is close to approaching zero.

Think of the i in the expression

0 + i

as the diameter of the earth, and 0 as the limit as the scaler of the versor gets closer and closer to zero. In other words approaches a right versor.

The unit vector i which is an infinitesimal segment of arc length one foot long and the vector of the versor which has the diameter of the earth both have a tensor of one.

(Three perpendicular diameters of the earth 'big i,j,k') and (two one foot long lines drawn on it, plus another one drawn straight up 'little i,k,k) have another interesting relationship, in that they both have the same multiplication table. Classical texts often introduce the multiplication table early on and tend to use the term quadrantal versor early, before the concept of a versor that can rotate something through an angle of other than 90 degrees is introduced.

After some important identities are proven the term quadrantal versor is used less often.

In must be constantly born in mind however that a vector and a versor are two different entities, and their laws of addition are different.

Radial quotient

The ratio of two vectors of equal length is called a radial quotient or a radial. A versor may also be viewed as the quotient of two vectors which are equal in length. In this case the arc can be visualized as the arc connecting the two vectors when they are placed tail to tail. In this representation the plane of the versor is the plane of the two vectors and the axis of the versor is a unit vector perpendicular to the plane. If the two equal length vectors in the quotient are at right angles to each other, then the radial quaternion is called a right radial quotient. An important property of right radial quotients is that their square is always equal to negative unity.

Degenerate forms

Two special degenerate versor cases, called the unit-scalars These two scalars, negative and positive unity can be thought of as scalar quaternions. These two scalars are special limiting cases, corresponding to versors with angles approaching either zero or π. Zero and π are then two special scalar points of singularity.

Unlike other versors, these two cannot be represented by a unique arc. The arc of one is a single point, and minus one can be represented by an infinite number of arcs, because there are an infinite number of shortest lines between antipodal points of a sphere.

Nonversor

The scalar number One was sometimes called the nonversor When a vector and the nonversor are multiplied the effect is not turning, hence the name nonversor.

One important formula for the nonversor was kji = +1.

It meant that this act of three successive versions or triple version, taken in this order, have the effect of neutralizing each other.

Hence kji represents the same operation as 1.

For any vector β,

kjiβ = β

Inversor

The scalar minus one was sometimes called the inversor.

The inversor, when multiplied with vector has the effect of reversing the direction of the vector.

The inversor can be thought of as an act of triple version.

ijk = -1

Hence ijk represents the same operation as −1

For any vector β,

ijkβ = −1β = −β

Quadrantal versor

A quadrantal versor has the effect of rotating a vector perpendicular to it by 90 degrees. Hence i × j = k. Here i represents an operator on j rotating it by 90 degrees. Using i as an operator again, i × k = −j. Classical notation viewed this as i operating on k to produce another rotation of 90 degrees. Note the logical consistency here: if it were true that i × (i × j) = −k, then it should also be true that (i × i) × j = −k and so i × i must equal minus one.

In multiplication, Minus one was called an inversor, having the effect on any vector of reversing it by 180 degrees to point in the opposite direction. Classical reasoning was that two successive rotations of 90 degrees in the same plane should produce the same effect as one rotation of 180 degrees. Quadrantal versors were therefore called semi-inversors. Quadrantal versors have a zero scalar component since the scalar component of a versor is the cosine of the angle of the versor.

There is another interpretation, that a right versor is some type of degenerate case. Recall that a versor is a kind of quaternion and is a four dimensional entity, where as a vector is a creature of pure dimension and is therefore a three dimensional entity. A quadrential versor, is introduced with out a scaler in front of it.

As the angle of a versor changes and it sweeps across its arch length, it approaches a limit where it suddenly becomes a different kind of decidedly vector like quantity.

Then after passing through 90 degrees the scalar of the versor becomes negative. In classical quaternion thinking positive scalars generally represent a positive quantity of time, and the notion of going from the present to the future, whereas negative scalars represent the notion of the past, or the notion when paired with a vector of places that exist in the past. This sudden flip is an important singularity.

Quaternion

Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space; or, more simply, as the quotient of two vectors. A quaternion which could be represented as the sum of a vector and a scalar.

A quaternion could be decomposed into a scalar and a vector, or into a tensor and a versor.

Every quaternion can be decomposed into a scalar and a vector.

$q=\mathbf {S} (q)+\mathbf {V} (q)\,$

These two operations S and V are called "take the Scalar of" and "take the vector of" a quaternion. The vector part of a quaternion is also called the right part.

In abridged notation, parentheses are not required and were not normally used. In the above expression Vq and Sq could be written without ambiguity. The operation of taking the vector of a quaternion took priority over the operation of raising to a power, unless a dot was placed between the operation and the rest of the expression, as in the relations below.

$\mathbf {V} q^{2}=(\mathbf {V} q)^{2}$

$\mathbf {V} .q^{2}=\mathbf {V} (q^{2})\,$

The operations "take the tensor of" and "take the versor of" could then decompose the vector of a quaternion V(q) further into a tensor and a unit vector. Like all vectors, this unit vector has the property that its square equals the scalar minus one.

The first of these operations would be written s=T(v). The second operation, taking the versor of the vector, returns a unit vector u=U(v). A unit vector is also a special type of versor with an angle of 90 degrees; hence a unit vector can rightfully be called a special type of versor called a right versor.

$\mathbf {V} q=(\mathbf {T} .\mathbf {V} q)(\mathbf {U} .\mathbf {V} q)\,$

Right quaternion

A basically right quaternion is quaternion whose scalar component is zero, $S(q)=0$. The angle of a right quaternion is 90 degrees. A right quaternion can also be thought of as a vector plus a zero scalar. More technically right quaternion can be thought of as either having a scaler part equal to or more correctly in some contexts approaching the limit of zero.

Right quaternions may be put in what was called the standard trinomial form. For example, if Q is a right quaternion, it may be written as:

$Q=xi+yj+zk\,$

It can be proven that every vector function is a function of a right quaternion.

The concept of the right quaternion is closely related to that of a quadrantial versor, and technically these are types of right quaternions. The quantities i,j,k and the term Quadrantial Versor are introduced early on in Hamilton's text and lectures, as he introduces the idea of multiplication, and his formulas i × j = k. Later on after he has offered much more precise definitions of concepts with small but important differences he used those terms, and drops the use of quadrantial versor. The notion of a quadrantial versor is broad umbrella term that covers a wide range of topics, that Hamilton uses as a device until he can create precise definitions.

What Hamilton is working up to is proof that 90 degree arc lengths, and the basis vectors used in vector addition have an important relationship, in that they both have the same multiplication table and are both a species of Quadrantial versor. These distinctly different quantities also have important differences and hence a precise vocabulary is needed to discuss them in the level of detail that was undertaken in the classical era.

A right versor is a another special kind of right quaternion. A like all right quaternions a right versor has a scaler part equal to zero. The tensor of a right versor is equal to one. Since a versor is a special kind of quaternion like all quaternions it can be decomposed into a scaler and a vector. A right versor is completely different from a vector because since a right versor has a scaler associated with it, it defines an angle, and can be viewed as a 90 degree segment of arch length. A versor and a vector have different rules of addition, because great circular arc lengths on spheres add differently than straight lines. If i is being used in the context of a right versor, i + i = -1, because the successive application of two 90 degree rotations has the effect of reversing the direction of a vector. If i is being used as a basis vector for a rectangular coordinate system, then i + i = 2i. The vector of a right versor The vector of a versor is then once again an element of pure distance not associated with time. A vector can be viewed as an infinitesimal segment of arc length. In this context a vector can be viewed as the vector part of a versor with an angle that approaches but never actually reaches zero. This corresponds to the notion that infinitesimal rotations add like vectors but finite rotations add by a different law, the law of arc length addition. Another limiting case is that of adding a vector with a tensor approaching the limit of zero to a scaler. This then is a scaler quaternion, since it contains both a scaler and a vector approaching the limit of zero. Like all quaternions a scaler quaternion can be divided into a vector and a scaler, and the vector can then be divided into a tensor and a unit vector having the same direction as the infinitesimal vector. The concept of the degenerate versors corresponds to the idea that negative one and positive one are special limiting cases that may be approached by but never reached by a versor because the angle of a versor is defined in the more precise classical texts as being greater than zero, and less than 180 degrees.

Geometrically real and geometrically imaginary numbers

In classical quaternion literature the equation

$q^{2}=-1\,$

was thought to have infinitely many solutions that were called geometrically real. These solutions are the unit vectors that form the surface of a unit sphere.

The term geometrically real roots of the above equation refers to quantities that can be written as a linear combination of the i,j and k, with the condition the sum of the squares of the coefficients of the expression add up to one. Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real ones. Given the existence of the imaginary scalar a number of expressions can be written and given proper names. All of these were part of Hamilton's original quaternion calculus.

$q+q'{\sqrt {-1}}$

where q and q' were real quaternions, and the square root of minus one was understood to be the imaginary of ordinary algebra, and called an imaginary or symbolical roots and not a geometrically real vector quantity.

Imaginary scalar

Geometrically Imaginary quantities are additional roots of the above equation of a purely symbolic nature. In article 214 or elements Hamilton proves that if there is an i j and k there also has to be another quantity h which is an imaginary scalar, which he observes should have already occurred to anyone who hard read the preceding articles with attention.Article 149 of elements of quaternions is an important article about Geometrically Imaginary numbers and includes a footnote introducing the term biquaterion. The term imaginary of ordinary algebra and scalar imaginary are sometimes used to refer to these geometrically imaginary quantities.

Geometrically Imaginary' roots to an equation were interpreted in classical thinking as geometrically impossible situations. Article 214 of elements of quaternions explores the example of the equation of a line and a circle that do not intersect, as being indicated by the equation having only a geometrically imaginary root.

In Hamilton's later writings he proposed using the letter h do denote the imaginary scalar

Bi-vector

Article 214 also defines a bivector as the product of a vector and the imaginary of ordnary algebra.

Biquaternion

Main article: Biquaternion

A Biquaternion is by definition the quotient of a bivector and a vector. It can also be written in this same form.

Other double quaternions

Hamilton invented the term associative to distinguish between the both commutative and associative imaginary scalar, and four other possible roots of negative unity. These he suggested should be given the designations L M N and O, and they are discussed very briefly in appendix B of Lectures on Quaternions, and in private letters however non-associative roots of minus one do not appear in Elements of Quaternions. Hamilton's life ended before he ever had a chance to work on these strange entities, they are a bow for another Ulysses

Four operations

Four operations are of fundamental importance in quaternion notation.

$+-\div \times$

In particular it is important to understand that there is a single operation of multiplication, a single operation of division, and a single operations of addition and subtraction. This single multiplication operator can operate on any of the types of mathematical entities. Likewise every kind of entity can be can be divided, added or subtracted from any other type of entity. Understanding the meaning of the subtraction symbol is critical in quaternion theory, because it leads to an understanding of the concept of a vector.

Ordinal operators

The two ordinal operations in classical quaternion notation were addition and subtraction or + and -.

These marks are:

"...characteristics of synthesis and analysis of a state of progression, according as this state is 
considered as being derived from, or compared with, some other state of that progression."

Subtraction

Subtraction is a type of analysis called ordinal analysis

...let space be now regarded as the field of progression which is to be studied, 
and POINTS as states of that progression. ...I am lead to regard the word "Minus," or the mark -, 
in geometry, as the sign or characteristic of analysis of one geometric position (in space), as compared 
with another (such) position. The comparison of one mathematical point with another with a view
to the determination of what may be called their ordinal relation, or their relative position in space...

The first example of subtraction is to take the point A to represent the earth, and the point B to represent the sun, then an arrow drawn from A to B represents the act of moving or vection from A to B.

B - A

this represents the first example in Hamilton's lectures of a vector. In this case the act of traveling from the earth to the moon.

Addition

Addition is a type of analysis called ordinal synthesis.

Addition of vectors and scalars

Vectors and scalars can be added. When a vector is added to a scalar, a completely different entity, a quaternion is created.

A vector plus a scalar is always a quaternion even if the scalar is zero. If the scalar added to the vector is zero then the new quaternion produced is called a right quaternion. It has an angle characteristic of 90 degrees.

Cardinal operations

The two Cardinal operations in quaternion notation are geometric multiplication and geometric division and can be written:

$\div \,\times$

It is not required to learn the following more advanced terms in order to use division and multiplication.

Division is a kind of analysis called cardinal analysis.Multiplication is a kind of synthesis called cardinal synthesis

Division

Classical books on quaternions first introduce the quaternion as the ratio of two vectors. This was sometimes called a geometric fraction.

If OA and OB represent two vectors drawn from the origin O to two other points A and B, then the geometric fraction was written as

$OA:OB\,$

Alternately if the two vectors are represented by α and β the quotient was written as

$\alpha \div \beta$

or

${\frac {\alpha }{\beta }}$

Hamilton spends a great deal of time on the development of the concept of a vector and is already 110 pages into Elements of Quaternions before he even introduces the word quaternion. At the end of article 112 Hamilton reaches the important conclusion he has been working up to: "The quotient of two vectors is generally a quaternion".

Lectures on Quaternions also first introduces the concept of a quaternion as the quotient of two vectors, if

Logically and by way of definition

If ${\frac {\alpha }{\beta }}=q$

then ${q}\times {\beta }=\alpha .$.

Notice that the order of the variables is of great importance. If the order of q and β were to be reversed the result would not in general be α. This is because the product in Hamilton's calculus is not commutative. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, what they used to call an act of version and then changing the length of it, which is what used to be call an act of tension. Also by definition the quotient of two vectors is equal to the numerator times the reciprocal of the denominator. Since multiplication of vectors is not commutative, the order can not be changed in the following expression.

${\frac {\alpha }{\beta }}=\,{\alpha }\times {\frac {1}{\beta }}$

Again the order of the two quantities on the right hand side of the equation is an important part of the classical definition of division.

Hardy presents the definition of division in terms of pneumonic cancellation rules. "Canceling being performed by an upward right hand stroke".

Like wise, alpha and beta are vectors and if q is a quaternion such that

${\frac {\alpha }{\beta }}=q$

then $\alpha \beta ^{-1}=q\,$

and

${\frac {\alpha }{\beta }}.\beta =\alpha \beta ^{-1}.\beta =\beta$

Lectures on Quaternions provides the following important formula on canceling.

β÷α×α = β and q×α÷α = q

γ = (γ÷β)×(β÷α)×α

An important way to think of q is as an operator that changes β into α alpha, by first rotating it, what they used to call an act of version and then changing the length of it, which is what used to be call an act of tension.

γ÷α = (γ÷β)×(β÷α)

Division of the unit vectors i, j, k

The results of the using the division operator on i,j and k was as follows.

$ij=k\,$		${\frac {k}{j}}=i$
$jk=i\,$		${\frac {i}{k}}=j$
$ki=j\,$		${\frac {j}{i}}=k$
$ji=-k\,$		${\frac {-k}{i}}=j$
$kj=-i\,$		${\frac {-i}{j}}=k$
$ik=-j\,$		${\frac {-j}{k}}=i$
$i(-j)=-k\,$		${\frac {-k}{-j}}=i$
$i(-k)=j\,$		${\frac {j}{-k}}=i$
$k(-i)=-j\,$		${\frac {-j}{-i}}=k$
$k(-j)=i\,$		${\frac {i}{-j}}=k$
$j(-k)=-i\,$		${\frac {-i}{-k}}=j$
$j(-i)=k\,$		${\frac {k}{-i}}=j$

The reciprocal of a unit vector is the vector reversed.

${\frac {1}{i}}=i^{-1}=-i\,$

Because a unit vector and its reciprocal are parallel to each other but point in the opposite directions, product of a unit vector and its own reciprocal have a special case commutative property, which does not hold in general for other products of unit vectors, for example if a is any unit vector then:

${\frac {1}{a}}a=(-a)a=1=a(-a)=a{\frac {1}{a}}.$

However in the more general case involving more than one vector the commutative property does not hold. For example:

$i{\frac {k}{i}}$ ≠ ${\frac {k}{i}}i.$

This is because k/i is carefully defined as:

${\frac {k}{i}}=k{\frac {1}{i}}=ki^{-1}=k(-i)=-(ki)=-(j)=-j$.

So that:

$i{\frac {k}{i}}=i(-j)=-k$,

however

${\frac {k}{i}}i=(-j)i=-(ji)=-(-k)=k$

Division of two parallel vectors

While in general the quotient of two vectors is a quaternion, If α and β are two parallel vectors then the quotient of these two vectors is a scalar. For example if

$\alpha =ai\,$,

and $\beta =bi\,$ then

$\alpha \div \beta ={\frac {\alpha }{\beta }}={\frac {ai}{bi}}={\frac {a}{b}}$

Where a/b is a scalar.

Division of two non-parallel vectors

The quotient of two vectors is in general the quaternion:

$q={\frac {\alpha }{\beta }}$$={\frac {T\alpha }{T\beta }}(\cos \phi +\epsilon \sin \phi )$

Where α and β are two non-parallel vectors, φ is that angle between them, and e is a unit vector perpendicular to the plane of the vectors α and β, with its direction given by the standard right hand rule.

Multiplication

Classical quaternion notation system had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation.

Multiplication of a scalar and the vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.

Factor, Faciend and Factum

Factor x Faciend = Factum

When two quantities are multiplied the first quantity is called the factor and the second quantity is called the faciend and the result is called the factum.

Distributive

In the classical notation system, the operation of multiplication was distributive. Understanding this makes it simple to see why the product of two vectors in classical notation produced a quaternion.

$q=(ai+bj+ck)\times (ei+fj+gk)\,$

$q=ae({i}\times {i})+af({i}\times {j})+ag({i}\times {k})+be({j}\times {i})+bf({j}\times {j})+bg({j}\times {k})+ce({k}\times {i})+cf({k}\times {j})+cg({k}\times {k})$

Using the quaternion multiplication table we have:

$q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)\,$

Then collecting terms:

$q=-ae-bf-cg+(bg-cf)i+(ce-ag)j+(af-be)k\,$

The first three terms are a scalar.

Letting

$w=-ae-bf-cg\,$

$x=(bg-cf)\,$

$y=(ce-ag)\,$

$z=(af-be)\,$

So that the product of two vectors is a quaternion, and can be written in the form:

$q=w+xi+yj+zk\,$

Product of two right quaternion

The product of two Right Quaternions is generally a quaternion. Let α and β be the right quaternions that result from taking the vectors of two quaternions:

$\alpha =\mathbf {V} p$

$\beta =\mathbf {V} q$

Their product in general is then a new quaternion represented here by r. This product is not ambiguous because classical notation has only one product.

$r=\,\alpha \beta ;$

Like all quaternions r may now naturally be decomposed into its vector and scalar parts.

$r=\mathbf {S} r+\mathbf {V} r$

The terms on the right are called scalar of the product, and the vector of the product of two right quaternions.

Other operators in detail

Taking the scalar and vector of a quaternion

Two important operations in two the classical quaternion notation system were S(q) and V(q) which meant take the scalar part of, and take the imaginary part, what Hamilton called the vector part of the quaternion. Here S and V are operators acting on q. Parenthesis can be omitted in these kinds of expressions without ambiguity.

In the classical era this is what the notation looked like:

$q=\,\mathbf {S} q+\mathbf {V} q$

Here, q is a quaternion. Sq is the scalar of the quaternion while Vq is the vector of the quaternion.

Take the tensor of

The operation called take the tensor of is represented by the letter T. It returns a kind of number called a tensor. Parenthesis are normally not needed for these types of expressions.

The tensor of a positive scalar is the scalar itself. The tensor of a negative scalar is the scalar with out the negative sign. For example:

$\mathbf {T} (5)=5\,$

$T(-5)=5\,$

The tensor of a vector is by definition the length of the vector. For example if:

$\alpha =xi+yj+zk\,$

Then

$\mathbf {T} \alpha ={\sqrt {x^{2}+y^{2}+z^{2}}}$

The tensor of a unit vector is one. Since the versor of a vector, is a unit vector, the tensor of the versor of any vector what so ever is always equal to unity, in symbols:

$\mathbf {TU} \alpha =1$

A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these two vectors. In symbols:

$q={\frac {\alpha }{\beta }}.$

$\mathbf {T} q={\frac {\mathbf {T} \alpha }{\mathbf {T} \beta }}.$

From this definition it can be shown that a useful formula for the tensor of a quaternion is:

$\mathbf {T} q={\sqrt {w^{2}+x^{2}+y^{2}+z^{2}}}\,$

It can also be proven from this definition that another formula to obtain the tensor of a quaternion is from the common norm, defined as the product of a quaternion and its conjugate. The square root of the common norm of a quaternion has the property that it is equal to its tensor.

$\mathbf {T} q={\sqrt {qKq}}\,$

A useful identity is that the square of the tensor of a quaternion is equal to the tensor of the square of a quaternion, so that parenthesis may be omited.

$(\mathbf {T} q)^{2}=\mathbf {T} (q^{2})=\mathbf {T} q$

The tensors of conjugate quaternions are equal.

$\mathbf {TK} q=\mathbf {T} q$

If Q is a biquaternion then the operation of taking the tensor of a biquaternion returns a bitensor.

$\mathbf {T} Q=t+{\sqrt {-1}}t'$

Here t and t' are reals.

Decomposition into tensor and versor

Using the important pair of classical quaternion operations, take the tensor of and take the versor of a quaternion can be deconstructed into a tensor and versor:

$q=\mathbf {T} q.\mathbf {U} q\,$

Any vector may be decomposed into a tensor and a unit vector.

This can be written

$\alpha =\mathbf {T} \alpha \mathbf {U} \alpha$

A unit vector written

$\mathbf {U} \alpha$

can always be extracted from any vector.

The tensor of a unit vector is always equal to one.

$\mathbf {T} .\mathbf {U} \alpha =1$

In words the formula above says the tensor of the unit vector of alpha equals one.

An example of a unit vector is i. This symbol i represents a unit of pure direction. Multiplying this unit of pure direction by a tensor then can make it longer or shorter but can never changed its direction.

Taking the conjugate

The K(q) operator means, take the conjugate. The conjugate of a quaternion is another quaternion obtained by multiplying the vector part of the first quaternion by minus one.

If

$q=\,\mathbf {S} q+\mathbf {V} q$

then

$\mathbf {K} q=\mathbf {S} \,q-\mathbf {V} q$.

The expression

$r=\,\mathbf {K} q$,

means, assign the quaternion r the value of the conjugate of the quaternion q.

Axis and angle of a quaternion

Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π.

When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector pointing perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule. The angle is the angle between the two vectors.

In symbols,

$u=Ax.q\,$

$\theta =\angle q$

Reciprocal of a quaternion

If

$q={\frac {\alpha }{\beta }}$

then its reciprocal is defined as

${\frac {1}{q}}=q^{-1}={\frac {\beta }{\alpha }}$

The expression:

${q}\times {\alpha }\times {\frac {1}{q}}$

Has many important applications for example rotations, particularly when q is the special type of quaternion called a versor. A versor has an easy formula for its reciprocal.

${\frac {1}{(\mathbf {U} q)}}=\mathbf {S.U} q-\mathbf {V.U} q=\mathbf {K.U} q$

In words this says that the reciprocal of a versor is equal to its conjugate. The dots between operators show the order to take the operations in, and also help to indicate that S and U for example are two different operations rather than a single operation named SU.

Common norm

The product of a quaternion with its conjugate was called the common norm.

The operation of taking the common norm of a quaternion is represented with the letter N. By definition the common norm is the product of a quaternion with its conjugate. It can be proven that common norm is equal to the square of the tensor of a quaternion. However this proof does not constitute a definition. Hamilton gives an exact definition both the common norm and the tensor, which do not depend on each other. This norm was adopted as suggested from the theory of numbers however to quote Hamilton "they will not often be wanted". The tensor is generally of greater utility. The word norm does not appear at all in Lectures on Quaternions, and only appears twice in the table of contents of Elements of Quaternions.

In symbols:

$\mathbf {N} q=\,q\mathbf {K} q=\,(\mathbf {T} q)^{2}$

The common norm of a versor is always equal to positive unity.

$\mathbf {NU} q=\mathbf {U} q.\mathbf {KU} q=1\,$

See also

• Cayley–Dickson construction
• Octonions
• Frobenius theorem

Footnotes

References

• W.R. Hamilton (1853), Lectures on Quaternions, Dublin: Hodges and Smith
• W.R. Hamilton (1899), Elements of Quaternions, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.
• A.S. Hardy (1887), Elements of Quaternions
• P.G. Tait (1890), An Elementary Treatise on Quaternions, Cambridge: C.J. Clay and Sons
• Herbert Goldstein(1980), Classical Mechanics, 2nd edition, Library of congress catalog number QA805.G6 1980