The vector of a quaternion


The vector of a quaternion

In the 19th century, the vector of a quaternion written Vq was a well defined mathematical entity in the classical quaternion notation system. This article is written using classical nomenclature. In this article the word vector means the classical vector of a quaternion.

By definition the square of every vector was equal to negative one. [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05230001;q1=vector;frm=frameset;view=image;seq=221;page=root;size=S Hamilton Lectures on Quaternions Lecture 3 Article 85 pg 81 1853] ]

Strictly speaking a vector in the classical quaternion system is an entity consisting of pure dimension.

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. [ [http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=tait+quaternion#PPA5,M1 Tait Elementary Treaties on quaternions page 4] ]

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 scaler 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.

=Decomposition of a vector into a tensor and a unit vector=

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

This can be written

α = TαUα

Is a unit vector. [ [http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA135,M1 Hamilton Elements pg 135] ] The tensor of a unit vector is always equal to one.

T.Uα

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

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.

=Addition of a vector and a scaler=

An important distinction is the difference between a vector and a vector plus a scaler. When a vector is added to a scaler, a completely different entity is created. This new entity is called a quaternion and represents a quantity with both a space and time component.

A quaternion can be used to represent different things, in some contexts it can represent an operator that rotates a vector and in other contexts it can represent a location in both space and time.

A vector plus a scaler is always a quaternion even if the scaler is zero. A vector plus a scaler is a complex number consisting of two distinct element.

If the scaler added to the vector is zero then the new quaternion produced is called a right quaternion. It has a angle characteristic of 90 degrees.

A versor is a special kind of right quaternion. Right versor has an angle of 90 degrees. This corresponds to the notion of a 90 degree rotation about an axis. The tensor of a versor is always equal to one. Since a right versor is a versor like all versors the tensor of a right versor is equal to one. The versor of a versor is given a special definition and indicates the extraction of a three dimensional unit vector from the four dimensional versor.

Product of a vector and another vector

The product of a vector and another vector is a quaternion.

Quotient of a vector and another vector

The quotient of a vector with another vector is a quaternion.

Classical quaternion notation had an operation called division. In fact most 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 quotent was writen as

α÷β orα/β

Hamilton 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". [ [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#PPA110,M1 Elements of Quaternions Article 112 page 110] ]

Order of factors is important

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

q = α÷β.

Logically and by way of definition then [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math&cc=math&idno=05140001&node=05140001%3A4&frm=frameset&view=image&seq=48 Hardy Elements of quaternions pg 32] ] [ [http://books.google.com/books?id=PJIKAAAAYAAJ&printsec=frontcover&dq=hamilton+lectures+on+quaternions#PRA1-PA37,M1 Hamilton Lectures on Quaternions page 37] ]

q ×β = α.

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. In other words by definition

α/β = (α) x (1/β)

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.

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

β/α = q then βα-1=q

and

β/α.α = βα-1.α=β [ [http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=tait+quaternion#PPA32,M1 Tait Treaties on Quaternions] ]

Rules for Canceling

Hardy [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05140001;node=05140001%3A4;frm=frameset;view=image;seq=60;page=root;size=S Hardy Elements of Quaternions pg 40] ] presents the definition of division in terms of pneumonic cancelation rules. "Canceling being performed by an upward right hand stroke".

Lectures on Quaternions provides the following important formula on canceling.

β÷α×α = β and q×α÷α = q [ [http://books.google.com/books?id=PJIKAAAAYAAJ&printsec=frontcover&dq=hamilton+lectures+on+quaternions#PRA1-PA38,M1 Hamilton Lectures On Quaternions pg 38] ]

γ = (γ÷β)×(β÷α)×α [ [http://books.google.com/books?id=PJIKAAAAYAAJ&printsec=frontcover&dq=hamilton+lectures+on+quaternions#PRA1-PA41,M1 Hamilton Lectures on quaternions page 41] ]

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) [ [http://books.google.com/books?id=PJIKAAAAYAAJ&printsec=frontcover&dq=hamilton+lectures+on+quaternions#PRA1-PA42,M1 Hamilton Lectures on quaternions pg 42] ]

Division of basis vectors

The results of the using the division operator on i,j and k was as follows. [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05140001;node=05140001%3A4;frm=frameset;view=image;seq=56;page=root;size=S Elements of Quaternions page 40 1887] ]

frac{k}{j}=ibecause k= ij
frac{i}{k}=jbecause i=jk
frac{j}{i}=kbecause j=ki
frac{-k}{i}=j
frac{-i}{j}=k
frac{-j}{k}=i
frac{-k}{-j}=i
frac{j}{-k}=i
frac{-j}{-i}=k
frac{i}{-j}=k
frac{-i}{-k}=j
frac{k}{-i}=j
γ

=The Vector as an element of a quaternion=

Every quaternion contains a vector.

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

q = S(q) + V(q)

These two operations S and V were called take the Scaler of and take the vector of a quaternion. Vq and Sq could be written with out ambiguity.

=A vectors is a limiting case of a right quaternion=

Right Quaternions

A right quaternion is zero plus a vector. A right quaternion has an angle of 90 degrees. A right quaternion has a scaler part equal to or more correctly in some contexts approaching the limit of zero.

Quadrantial Versors, Unit Vectors, and Right Versors.

Hamilton offers very precise definitions for each of these entities, to distinguish between them. All of them are very different concepts with different definitions.

Quadrantial Versors

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 x 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.

Unit vectors used as basis vectors in rectangular coordinates

i,j and k can also be used as the basis vectors of a rectangular coordinate system. In this context they work exactly like the vectors of other notation systems, when it comes addition.

19th century books on quaternion geometry normally devote an entire early chapter to vector addition.

Right Versors

A right versor is a 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.

The vector of a right versor

An important point to remember is that the unit vector of a right versor is much longer than a different but related unit vector corresponding an infinitesimal unit tangent drawn along the arc defined by the same right versor.

Three dimensional rectangular unit vectors as Hamilton proved [ [http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA145&dq=vector-arcs&lr=#PPA142,M1 Hamilton Elements of quaternions section 9 about vector arcs starting on pg 142] ] 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.

Vector-Arc

As an example give the unit vector that defines an infinitesimal unit of arc [ [http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA145&dq=vector-arcs&lr=#PPA142,M1 Hamilton Elements of quaternions section 9 about vector arcs starting on pg 142] ] 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 i,j,k of a vectors and a versors have an important relationship

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 perpindicular 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. [ [http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA145&dq=vector-arcs&lr=#PPA157,M1 See Hamilton Elements of quaternions pg 157 section 10 titled On the system of three right versors in the rectangular plane and the laws of i,j,k] ] 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.

Quadrential versor as degenerate case.

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 number, a vector. Then after passing through 90 degrees the scaler of the versor becomes negative. In classical quaternion thinking positive scalers generally represent a positive quantity of time, and the notion of going from the present to the future and negative scalers represent the notion of the past, or the notion when paired with a vector of places that exist in the past.

=One and minus one are unit-scalars and degenerate versors=

In addition to the first case, the special angle of 90 degrees there are two other special degenerate versor cases, called the unit-scalars [ [http://books.google.com/books?id=fIRAAAAAIAAJ&pg=PA117&dq=quaternion&lr=#PPA130,M1 Hamilton Elements of Quaternions Article 147 pg 130] ]

These are the angle zero, where the versor, which is a quaternion suddenly becomes the scaler unity written as 1.

The third case is where the angle becomes 180 degrees and the versor degenerates to negative unity -1.

One the Nonversor

The number plus one written +1 was normally called unity, howeverclassical quaternion thinking the number positive one could also technically be called the nonversor [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=nonversor;rgn=full%20text;idno=05230001;didno=05230001;view=image;seq=380 Hamilton Lectures on quaternions pg 240] ] [ [http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math&cc=math&idno=05230001&q1=nonversor&frm=frameset&view=image&seq=194 non-versor on pg 54 Lectures on Quaternions] ] .

The nonversor +1 was not a vector or a versor but was rather considered to be a scaler.

The nonversor could be thought of as a scaler quaternion, in other words a quaternion consisting of just a scaler with no vector part. In this context the nonversor was a quaternion with an angle of zero.

The nonversor is one of two degenerate versors and was an important singularity.

A versor by definition has an angle greater than 0 and less than 180 degrees, but the nonversor could be thought of as a limiting case as the angle of a versor approached zero hence it was called a degenerate versor.

The 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 = 1β = β

Minus one the Inversor

The another degenerate limiting case in the classical quaternion system of thinking and notation was that of the inversor. The inversor can be thought of as a special limiting case as the angle of a versor approaches 180 degrees.

The inversor was a scaler.

The inversor was represented by the symbol -1.

The square of the inversor was positive one.

Since the square of the inversor and the nonversor were positive one, these were not considered to be vectors which by definition have a square of negative one.

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

Footnotes

An [http://books.google.com/books?id=CGZLAAAAMAAJ Elementary Treatise on Quaternions] By Peter Guthrie TaitPublished by University Press, 1890Original from the University of MichiganDigitized Oct 3, 2007422 pages


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