Classical Hamiltonian quaternions

From Wikipedia, the free encyclopedia

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 describes Hamilton's original treatment of quaternions, using his own notation and definition of terms. This treatment is more geometric than the modern treatment, which emphasizes the algebraic properties of quaternions. Mathematically speaking, the quaternions discussed in this article are the same quaternions that are used in almost all modern applications.

[edit] Classical elements of a quaternion

There are a number of different types of entities used in Hamilton's quaternion calculus. The four most important kinds of geometric entities are tensors, vectors, scalars and versors. Each of these four 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, can be split into a vector and a scalar, as well as into a tensor and a versor.

Hamilton introduces various operators which act on thes geometric entities. The operators 'take the tensor of', 'take the versor of', 'take the scalar of', 'take the vector of' and 'take the conjugate of' are denoted by T,U,S,V and K.

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

[edit] 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"^[1] or because they represent the "comparison of positions upon one common scale".^[2]Hamilton regarded ordinary scalar algebra the science of pure time.^[3]

[edit] Vector

Hamilton defined a vector as "a right line ... having not only length but also direction".^[4] Hamilton derived the word vector from the Latin vehere, to carry.^[5]

Hamilton's conceived a vector as the "difference of its two extreme points.^[6] For Hamilton, a vector was always a three dimensional entity, having three co-ordinates relative to any given co-ordinate system, including but not limited to both polar or rectangular co-ordinates.^[7] He therefore referred to vectors as "triplets".

Hamilton defined addition of vectors in geometric terms, by placing the origin of the second vector at the end of the first.^[8] He went on to define subtraction of vectors.

By adding a vector to itself multiple times, he defined multiplication of a vector by a integer, then extended this to division by an integer, and multiplication (and division) of a vector by a rational number. Finally, by taking limits, he defined the result of multiplying a vector α by any scalar x as a vector β with the same direction as α if x is positive; the opposite direction to α if x is positive; and a length that is |x| times the length of α.^[9]

The quotient of two parallel or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.^[10]

The square of every vector is a negative scalar.^[11]

[edit] Unit vector

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

[edit] 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.

In Hamilton's usage, a tensor is defined as a positive numerical quantity, or more properly signless, number.^[12]^[13]^[14]A tensor can be thought of as a positive scalar^[15] The "tensor" can be thought of as representing a "stretching factor."^[16]

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 algebraic 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.

[edit] Versor

Main article: versor

A versor is a quaternion with a tensor of 1. Alternatively, a versor can be defined as the quotient of two vectors of equal length.^[17]^[18] Each quaternion is equal to a versor multiplied by the tensor of the quaternion. Denoting the versor of a quaternion by

$\mathbf{U}q$

we have

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

In general a Versor can be associated with a plane, an axis and an angle.^[19]

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.

[edit] Vector arc

A versor can also in general be represented by a unique great circle arc, called a vector arc.^[20]^[21] 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.

[edit] 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.

[edit] Radial quotient

The ratio of two vectors of equal length is called a radial quotient or a radial.^[22] 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.^[23]

[edit] Degenerate forms

Two special degenerate versor cases, called the unit-scalars^[24] 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.

[edit] Nonversor

The scalar number One was sometimes called the nonversor^[25]^[26] 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

[edit] Inversor

The scalar minus one was sometimes called the inversor.^[27]

The inversor, when multiplied with vector has the effect of reversing the direction of the vector.^[28]

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

[edit] 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.^[29] 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) = −j, then it should also be true that (i × i) × j = −j and so i × i must equal minus one.^[30]

[edit] Quaternion

Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space;^[31] or, more simply, as the quotient of two vectors.^[32] 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.^[33]

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.

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

$\mathbf{V}.q^2=\mathbf{V}(q^2)\,$ ^[34]

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.

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

[edit] 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 scalar 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:

$Q = xi + yj + zk\,$ ^[35]

[edit] 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^[36] and not a geometrically real vector quantity.

[edit] 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.^[37]Article 149 of elements of quaternions is an important article about Geometrically Imaginary numbers and includes a footnote introducing the term biquaterion.^[38] 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.^[39]

In Hamilton's later writings he proposed using the letter h do denote the imaginary scalar^[40]^[41]^[42]

[edit] Bi-scalar

[edit] Bi-vector

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

[edit] 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.

[edit] 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^[43]

[edit] Four operations

Four operations are of fundamental importance in quaternion notation.^[44]

$+ - \div \times$

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.

[edit] 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."^[45]

[edit] Subtraction

Subtraction is a type of analysis called ordinal analysis^[46]

...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...^[47]

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.^[48]^[49]

[edit] Addition

Addition is a type of analysis called ordinal synthesis.^[50]

[edit] 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.

[edit] Cardinal operations

The two Cardinal operations^[51] 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.^[52]Multiplication is a kind of synthesis called cardinal synthesis^[53]

[edit] 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".^[54]

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

Logically and by way of definition^[55]^[56]

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^[57] 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

$\frac{\alpha}{\beta}.\beta = \alpha\beta^{-1}.\beta=\beta$ ^[58]

Lectures on Quaternions provides the following important formula on canceling.

β÷α×α = β and q×α÷α = q^[59]

γ = (γ÷β)×(β÷α)×α^[60]

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)^[61]

[edit] Division of the unit vectors i, j, k

The results of the using the division operator on i,j and k was as follows.^[62]

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

The reciprocal of a unit vector is the vector reversed.^[63]

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

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:^[64]

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

However in the more general case involving more than one vector the commutative property does not hold.^[65] 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

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

[edit] 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.^[66]

[edit] 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.^[67]

[edit] 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.

[edit] Factor, Faciend and Factum

Factor x Faciend = Factum^[68]

When two quantities are multiplied the first quantity is called the factor^[69] and the second quantity is called the faciend and the result is called the factum.

[edit] 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:

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

[edit] 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^[70] of two right quaternions.

[edit] Other operators in detail

[edit] 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.

[edit] 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.

[edit] 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:

$\mathbf{TU}\alpha = 1$ ^[71]

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:

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

$\mathbf{T}q = \frac{\mathbf{T}\alpha}{\mathbf{T}\beta}.$ ^[72]

From this definition it can be shown that a useful formula for the tensor of a quaternion is:^[73]

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

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 omitted.^[74]

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

The tensors of conjugate quaternions are equal.^[75]

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

If Q is a biquaternion then the operation of taking the tensor of a biquaternion returns a bitensor.^[76]

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

Here t and t' are reals.

[edit] 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.^[77]^[78]

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.^[79] The angle is the angle between the two vectors.

In symbols,

$u = Ax.q\,$

$\theta = \angle q$

[edit] 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^[80]^[81] for example rotations, particularly when q is the special type of quaternion called a versor. A versor has an easy formula for its reciprocal.^[82]

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

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.

[edit] Common norm

The product of a quaternion with its conjugate was called the common norm.^[83]

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^[84]^[85] 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.^[86]

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

[edit] See also

[edit] Footnotes

[edit] References

W.R. Hamilton (1853), Lectures on Quaternions, Dublin: Hodges and Smith
W.R. Hamilton (1866), 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

[0] Hamilton, in the Philosophical magazine, as cited in the OED.

[1] Hamilton (1866) Book I Chapter II Article 17

[2] Hamilton 1853, pg 2 paragraph 3 of introduction. Refers to his early article Algebra as the Science of pure time.

[3] Hamilton (1866) Book I Chapter I Article 1

[4] Hamilton (1853) Lecture I Article 15, introduction of term vector, from vehere

[5] Hamilton (1866) Book I Chapter I Article 1

[6] Hamilton (1853) Lecture I Article 17 vector is natural triplet

[7] Hamilton (1866) Book I Chapter I Article 6

[8] Hamilton (1866) Book I Chapter I Article 15

[9] Hamilton (1866) Book I Chapter II Article 19

[10] Hamilton 1853 Lecture 3 Article 84 pg.81

[11] "Hamilton 1853 pg 57". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=tensor+definition+positive+properly+new+word#PRA1-PA57,M1.

[12] "Hardy 1881 pg 5". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=positive+tensor+strictly+speaking+number+without+sign&as_brr=1#PPA5,M1.

[13] "Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number". http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=Hamilton+positive+signless+quotients#PPA31,M1.

[14] "Hamilton 1989 pg 165, refers to a tensor as a positive scalar.". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=tensor+scalar+positive&lr=#PPA165,M1.

[15] "Tait (1890), pg 32". http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=Hamilton+positive+signless#PPA31,M1.

[16] "Hamilton 1898 section 8 pg 133 art 151 On the versor of a quaternion or a vector and some general formula of transformation". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=versor+radial+quotient&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA133,M1.

[17] "Hamilton (1899), art 156 pg 135, introduction of term versor". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA135,M1.

[18] "Hamilton (1899), Section 8 article 151 pg 133". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=elements+quaternions&printsec=frontcover&source=web&ots=DCbK_TcfUM&sig=D_8WVDm2V5xvWykppt0Cml3RCrg&sa=X&oi=book_result&resnum=10&ct=result#PPA133,M1.

[19] "Hamilton 1898 section 9 art 162 pg 142 Vector Arcs considered as representative of verosrs of quaternions". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=vector-arcs+representatives+versors+great+circle&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA142,M1.

[20] "Hardy (1881), art. 49 pg 71-72". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=representation+versors+spherical+vector+arcs&as_brr=1#PPA71,M1.

[21] "Hamilton (1899), Article 146 pg 129". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA129,M1.

[22] "Hamilton 1898 pg. 130, art. 148 The square of a right radial is negative unity". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=148+square+radial+negative+unity+general+property&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA130,M1.

[23] "Hamilton Elements of Quaternions Article 147 pg 130". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=147+Unit-Scalars&lr=#PPA130,M1.

[24] "Hamilton (1853), page 240". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=nonversor&as_brr=1#PRA1-PA240,M1.

[25] "Hamilton 1853 Pg 54-55 explains non-version". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=non-version&as_brr=1#PRA1-PA55,M1.

[26] "Hamilton (1899), art. 153 pg 54". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA134,M1.

[27] "See Hamilton (1853), Art. 60 starting on page 53 rule of the signs". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=60+rule+signs&as_brr=1#PRA1-PA53,M1.

[28] "Hardy (1881), page 41". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=quadrantal+versors+turn+line+90&as_brr=1#PPA41,M1.

[29] "Hamilton (1853), pg 70". http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=Quadrantal%20versor;rgn=full%20text;idno=05230001;didno=05230001;view=image;seq=210.

[30] "Hamilton 1853 pg. 60". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=quaternion+quotient+lines+tridimensional+space+time&as_brr=1#PPA60,M1.

[31] "Hardy 1881 pg. 32". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=quotient+two+vectors+called+quaternion&as_brr=1#PPA32,M1.

[32] "See Elements of Quaternions Section 13 starting on page 190". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=imaginary+roots&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA190,M1.

[33] "See Elements of Quaternions bottom of page 195". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=abridgment&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA195,M1.

[34] "Hamilton (1899), Section 14 article 221 on page 233". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA233,M1.

[35] "See Elements of Quaternions Articles 256 and 257". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=Symbolical+imaginary+roots&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA275,M1.

[36] "Hamilton Elements article 214 infamous remark...as would already have occured to anyone who had read the preceding articles with attention". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=214+read+recent+articles+attention&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA218,M1.

[37] "Elements of Quaternions Article 149". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=Geometrical+Imaginaries+Biquaternions&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA131,M1.

[38] "See elements of quaternions article 214". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=imaginary+roots&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA217,M1.

[39] "Hamilton Elements of Quatenions pg 276 Example of h notation for imaginary scalar". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=imaginary+biquaternion+denote&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA276,M1.

[40] "Hamilton Elements Article 274 pg 300 Example of use of h notation". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=imaginary+biquaternion+denote&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA300,M1.

[41] "Hamilton Elements article 274 pg. 300 Example of h denoting imaginary of ordinary algebra". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=imaginary+biquaternion+denote&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA300,M1.

[42] "Bow for another Ulysses is a famous remark made by Hamilton's son, in the preface of Elements 1866". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=bows+Ulysses+reserved&lr=#PPP12,M1.

[43] "Hamilton 1853 pg 4". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=four+operations#PRA1-PA4,M1.

[44] Hamilton 1853 art 5 pg 4 -5

[45] "Hamilton pg 33". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=subtraction+ordinal+analysis#PRA1-PA33,M1.

[46] Hamilton 1853 pg 5-6

[47] see Hamilton 1853 pg 8-15

[48] Hamilton 1853 pg 15 introduction of the term vector as the difference between two points.

[49] Hamilton 1853 pg.19 Hamilton associates plus sign with ordinal synthesis

[50] "Hamilton (1853), pg 35, Hamilton first introduces cardinal operations". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=cardinal+operations#PRA1-PA35,M1.

[51] "Hamilton 1953 pg.36 Division defined as cardinal analysis". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=division+analysis+analyzand+analyzer#PRA1-PA36,M1.

[52] "Hamilton 1853 pg 37". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=multiplication+cardinal+synthesis#PRA1-PA37,M1.

[53] "Hamilton (1899), Article 112 page 110". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quotient+two+vectors+quaternion&lr=#PPA110,M1.

[54] "Hardy (1881), pg 32". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=definition+quotient&as_brr=1#PPA32,M1.

[55] Hamilton Lectures on Quaternions page 37

[56] "Hardy (1881), pg 46". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=right+handed+stroke&as_brr=1#PPA46,M1.

[57] Tait Treaties on Quaternions

[58] Hamilton Lectures On Quaternions pg 38

[59] Hamilton Lectures on quaternions page 41

[60] Hamilton Lectures on quaternions pg 42

[61] "Hardy (1881), page 40-41". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=definition&as_brr=1#PPA40,M1.

[62] "Hardy 1887 pg 45 formula 29". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=reciprocal+unit+vectort&as_brr=1#PPA45,M1.

[63] "Hardy 1887 pg 45 formula 30". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=reciprocal+unit+vectort&as_brr=1#PPA45,M1.

[64] "Hardy 1887 pg 46". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=appear+correct+not+true&as_brr=1#PPA46,M1.

[65] Elements of Quaternions, book one.

[66] "Hardy (1881), pg 39 article 25". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=relation+symbolic+notation+quotient+vectors&as_brr=1#PPA39,M1.

[67] Hamilton 1853 pg. 27 explains Factor Faciend and Factum

[68] Hamilton 1898 section 103

[69] "Hardy (1887) scalar of the product vector of the product defined, pg 57". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=scalar+product+vector&as_brr=1#PPA57,M1.

[70] Hamilton 1898 pg164 Tensor of the versor of a vector is unity.

[71] "See all of section 11 Elements of Quaternions Hamilton 1898". http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=tensor+vector+quaternion&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA162,M1.

[72] "Hardy (1881), pg 65". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=Tq&as_brr=1#PPA65,M1.

[73] Hamilton 1898 pg 169 art 190 Tensor of the square is the square of the tensor

[74] Hamilton 1898 pg 167 art. 187 equation 12 Tensors of conjugate quaternions are equal

[75] "Hamilton 1853 pg 655-666 Introduction of the term bitensor in conjunction with biquaternion". http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=bitensor+biquaternion#PRA1-PA665,M1.

[76] "Hamilton (1853), pg 164, art 148". http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05230001;node=05230001%3A4;frm=frameset;view=image;seq=304;page=root;size=S.

[77] "Hamilton (1899), pg 118". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA118,M1.

[78] "Hamilton (1899), pg 117". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA117,M1.

[79] See Goldstein (1980) Chapter 7 for the same function written in matrix notation

[80] "Lorentz Transforms Hamilton (1853), pg 268 1853". http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05230001;q1=vector;frm=frameset;view=image;seq=408;page=root;size=S.

[81] "Hardy (1881), pg 71". http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=tensor+versor&as_brr=1#PPA34,M1.

[82] "Hamilton (1899), pg 128 -129". http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA128,M1.

[83] "See foot note at bottom of page, were word proven is highlighted.". http://books.google.com/books?id=fIRAAAAAIAAJ&dq=proved+Tensor+greater+utility+theory+numbers+not&printsec=frontcover&source=bn&hl=en&ei=sqyxSdbSDJmktQOkkKR8&sa=X&oi=book_result&resnum=5&ct=result#PPA129,M1.

[84] See Hamilton 1898 pg. 169 art. 190 for proof of relationship between tensor and common norm

[85] "Hamilton 1899 pg 138". http://books.google.com/books?id=fIRAAAAAIAAJ&dq=elements+of+quaternions&printsec=frontcover&source=bn&hl=en&ei=sqyxSdbSDJmktQOkkKR8&sa=X&oi=book_result&resnum=5&ct=result#PPA138,M1.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

[85]

[86]