BIS 



QUATERNIONS 



Quaternions (or 'acts of four'), the name of 

 calculus nf neculiar power and generality in 

 by Sir William Kowan Hamilton (<|.v.) of 



Dublin. AB a geometry, it primarily concerns 

 it.-elt with the operations by which one directed 

 quantity or Vector (q.v.) is changed into another. 

 Such an operation is called a quaternion, for 

 reasons which will appear hereafter. From this 

 point of view alone we shall discuss it hen- \\ > 

 assume the law of vector addition, which asserts 

 that the vector or directed 

 line AC (see fig. I ) is equal 

 to the sum of the vectors 

 AH and HO or anv other 

 directed lines parallel and 

 equal to them. For ex- 

 ample, the resultant of two 



Fig. 1. 



velocities or cotenninal forces is a vector equal to 

 the vector sum of the components (see COMPOSI- 

 TION). Quantities which do not involve the idea 

 of direction or directedness are called Sca/ars ; such 

 are the quantities used in arithmetic and ordinary 

 algebra. Parallel vectors can all be represented as 

 scalar multiples of one another, or (better) of the 

 parallel vector whose length is unity. By the 

 latter representation, the scalar multiple gives the 

 length or tensor of the vector. Thus any vector a 

 may be factorised into ii - tensor and directed unit 

 part. This is HymholiHed by the equation a = Tal'a, 

 where T and U appear as selective symbols of 

 operation, separating out the length and direction 

 respectively. 



The oitefation which simply rotates a vector into 

 a new direction without clianging its length is a 

 particular kind of quaternion called a Versor. A 

 second application of this versor produces an extra 

 equal rotation in the same plane i.e. about the 

 same axis. With every versor, therefore, are 

 associated an axis having a definite direction and 

 an angle through which anv vector perpendicular 

 to this axis U rotated by tfie versor operating on 

 it A very important 

 case is the quadrantal 

 or right versor, which 

 turns a perpendicular 

 vector through a right 

 angle. Let i represent 

 the right versor whose 

 axis is perpendicular to 

 the plane of the paper. 

 Then (lig. 2) if /3 is any 

 vector in the plane of 



the paper, the quantity i/S = y gives a vector per- 

 pendicular to ft and to the axis of t. A second 

 operation gives 



i/S = */S = i-r = -/3, 



or symliolienlly i* = - 1. Thus the square of any 

 righ't versor is negative unity. It is easy to show 

 that MI, where n is a scalar, is an operator which 

 Btill turn* any appropriate vector through a right 

 angle, hut at tin- -ame time increases its tensor n 

 linn. Such nn operator is a quadrantal quater- 

 nion, whose tensor is n and versor t. A quaternion 

 can always lie factoriited into iU tensor and versor 



Now let Oi, 01 (fig. 3) be the axes of two right 

 versors i and I, making angle 8 with each other. 

 Describe the sphere of unit radius with () as 

 centre, and draw the vector OA or a |>crpen- 

 dieuUr to i and in the plane Oil. Draw OI! or 

 p perpendicular to i and a i.e. upward from the 

 plane of the paper ; and finally draw OC or y per- 

 pendicular to 1 and ft. Then fimt 10 = ft and 

 secondly lia = Ift = y ; so that Ii ( = yfa) is the 

 Tenor which rotates a into the |M>silion 7. This 

 veraor hat its axis parallel to Oil, and its anyle 

 equal to the complement of 9. Thus any 



versor can lie represented by the product of two 

 ri^-lit versors perpendicular to it and making with 

 each other the appropriate angle. If the two right 

 venom are themselves at riglit angles, their pro- 

 duct becomes the right versor perpendicular to 

 Ixith. We thus arrive at what is historically the 

 basis of quaternions viz. Hamilton's remark- 

 able system of mutually perpendicular right verson, 

 ijk. As operators (see fig. 4) they are connected 

 by the equations 



The special point to notice is the non-commutative 

 character of t 



l>eing the 

 The discovery 



the process of multiplication, ij not 

 same as ''. 

 the 



of 



equation i - i 

 6, 



e 

 on 



October 16, 1843, was 

 quickly followed by the 

 development of the whole 

 calculus of quaternions. 

 Now, if j and /. were 

 vectors instead of right 

 versors, the equation 

 ij = I: would still lie true 

 as an equation of opera- 

 tions. In fact, as is cap- 

 able of easy proof, right 

 versors obey the law of 

 vector addition ; and in 

 the identification of unit 



Fig. a 



vectors and right 

 versors, or more generally of vectors and right 

 quaternions, lies one of the great simplifications 

 of the calculus. Thus the operator (t +j) is a 

 right quaternion whose axis (see fig. 4) is along 

 the diagonal of the square of which and j are the 

 sides, and whose tensor is equal to the length of 

 this diagonal. 



The following conclusions are readily come to. 

 The square of every unit vector is negative unity ; 

 the product of two parallel vectors is minus the 

 product of their tensors ; the product of two per- 

 pendicular vectors is a third vector perpendicular 

 to both and having it tensor equal to the prcnluct 

 of the tensors of its factors ; the product of any 

 two unit vectors is in 

 general a versor ; the *J 



product of any two 

 vectors is a quater- 

 nion whose tensor is 

 the product of the 

 tensors, and whose 

 vcrsor is as mentioned 

 in the preceding sen- 

 tence. The quater- 

 nion aft transforms 

 /S~' into the vector o ; 

 and B~ l , hciug itself 

 that quaternion which 

 undoes the effect of 

 the right quaternion 



Ki. 4. 



ft, must also he a right quaternion i.e. a vector. 

 In fact, ft ' is always equal to a scalar multiple 

 of - 3. Hence the quaternion a/9 is the operator 

 which changes the vector 0~' into the vector o. 

 This operation involves four numbers: first, the 

 change of length; second, the angle through which 

 the one vector must be rotated so as to bring it 

 into parallel ism with the other; and third and 

 fourth, the two numbers necessary to fix the asjiect 

 of the plane in which the rotation takes place, or 

 the direction of the axis about which rotation takes 

 place. Thus a quaternion, in general, depends on 

 four numbers, whence the name. A vector or 

 quadrantal quaternion is a degenerate quaternion, 



