Let another vector, OH, be drawn in the plane LMQP, making the angle HOE= EOF, and of such a length that OF: OE:: OE: OH. Then we have, without undergoing any change in value. 43°. It follows that no two diplanar quaternions can be equal. For suppose = OB OD OF OG = Then = and conse ОА OC OE OE' quently OF OG, which is contrary to definition, since the two vectors have not similar directions. Conversely, if two quaternions are equal, they are coplanar. 44°. If q and q' are equal quaternions, so that then, by definition, Sq+VqSq' + Vq'; Sq Sq', and Vq= Vq'. More generally, if an equation involves any number of scalar and vector quantities, the sums of the scalars and of the vectors on either side are respectively equal. example, let Then, and x + ma + nẞ = y + z + ty. ma + nß = (say) 18, x + 18 = (y + z) + ty. But (x + 18) and {(y + z) + ty} are quaternions, 20°. S(x+18), or S (x + ma + nẞ) = S {(y + z) + ty}, therefore, x = y + z; ma + nß = ty. For F CHAPTER V THE VARIOUS KINDS OF QUATERNIONS 45°. Collinear Quaternions. Quaternions whose planes intersect in, or are parallel to, a common line are said to be Collinear. For example, the quaternions OB. OA and OD. OC, fig. 19, 42°, are collinear ; and OL. ON, OM. ON, fig. 18, 35°, are also collinear, whatever be the angles YZ, ZX, ZY. Since the versors of collinears are each perpendicular to the common vector, it follows that if q, q', q', &c., be collinear, Uq, Uq', Uq", &c., are coplanar; and the converse. Coplanar quaternions are always collinear (or can be made. so by sliding and rotation in the plane), but the converse is not true. Collinears are not always coplanar. 46°. Reciprocal Quaternions. The Reciprocal of a quaternion in the form of a fraction is obtained by interchanging its divisor and dividend vectors. either of two reciprocals is equal to unity divided by the other, and that the product of the two is positive unity. In symbols, if q and q' be reciprocal, Reciprocal quaternions have, obviously, a common plane and angle, reciprocal tensors, and opposite axes-rotation from a to ẞ being contrary to rotation from ẞ to a; or, € (3) The versors of reciprocals are reciprocal, and e-c being reciprocal, = Uq-1= and Uq-1UqUqUq−1 = 1; . . (4) 1 Uq' = or, the versor of the reciprocal is equal to the reciprocal of the versor. 47°. Opposite Quaternions. If any two opposite vectors, ẞ and — ß, be divided by any one common vector, a, the two unequal quotients thus formed, and -B, are called opposite quaternions. Accordingly, — q β a α В is the opposite of q. Since, 1° (i), the sum of any two opposite quaternions is zero, and their quotient is negative unity, B Opposite quaternions, fig. 20, have a common plane, equal tensors, supplementary angles, and opposite axes, (2) FIG. 20. Ta (cos + e sin 0) = Ta В β (3) 48°. Let OA, OB, fig. 21, be any two vectors. From O draw OB' = OB in the plane AOB, making ▲ AOB' = ▲ AOB ; and draw BB', cutting OA produced in A'. Let OB' = ß'. В B' (a). The unequal quotients, and are called Conjugate α α =Kq, read 'conjugate of q' Conjugate quaternions have a common plane, equal angles and tensors, and opposite axes: -ce The versors of conjugates are reciprocal, since ea and e are reciprocal, and the product of the versors is positive unity : From the foregoing it is evident that aß and ßa are conjugate quaternions. Taß (- cos 0 sin 0), - € Kaß ßa = Tẞa (— cos 0 + € sin 0), we evidently have = Hence, we have as general expressions for a quaternion and its conjugate, whence, q = Sq + Vq, Kq=Sq - Vq; q + Kq = 2Sq, . qKq= 2Vq. Similarly, if q, q in (5) degrades to a negative x, and (6) becomes scalar, say If / 9 S vanishes in (5), q degrades to a positive π . (11) (12) . (13) If, therefore, 49 = – Sq vanishes in (12), — q 2' degrades to a negative vector, say -y, and (13) becomes From (9), (10), (11), and (14), it is clear that, (1) The conjugate of a scalar is the scalar itself; (d). By adding and subtracting equations (5) and (6), it is seen that while the sum of a quaternion and its conjugate is a scalar, their difference is a vector. (e). The most important formulæ of the last three sections are collected here for facility of reference : 49°. Miscellaneous Theorems. (a). The reciprocal of the reciprocal, the opposite of the |