172 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
direction is zero : it follows therefore that u.u—0, v.v—0 ; also that 
u.u-\-u.v-\-v.u-\-v .v — O, 
u.v-\-v .u — 0, 
or V.U——U.V. 
It appears then that u.v and v.u are equivalent as regards magnitude but opposite 
in sign. 
(33.) Secondly, as regards uXv, may we put vXu—uXv} This question is 
determined by observing that the longitudinal effect of the translation of a magnitude 
at right angles to its direction is zero, as follows. 
Let u=moi, v=m!a!, a and a! being the directions (directed units), and m and m! the 
magnitudes of u and v. Then, by article 30, 
uXv=^mn{aX<^') and vXu=nin {a'. a). 
Now it is clear, from figure 18, that a-f-a' and a — a! are lines at right 
angles to each other ; therefore 
(a -{-«') X (a— a') = 0 = (a — a') X {a-\-o !!) ; 
therefore, omitting common terms, we find 
— aX a'-\-cc! Xu = aXoc! — a' Xoc, 
aXot! — a' X a. 
And thus it follows that 
uxv=vXu. 
Fig. 19. 
J&ft 
ri^ht l<^t 
I"! 
'iykt 
It appears then that uXv and vXu are equivalent as regards both magnitude and sign. 
(34.) Thus u.v is commutative with change of sign, while mX*^ is simply commuta- 
tive. The reason of the change of sign in the former may be 
easily interpreted as follows. An arrow has two distinct sides, 
which, for the sake of fixing ideas, I may call right and left, 
and which may be defined by supposing that I stand on the 
plane of the paper looking in the direction of the arrow. Now, 
referring to fig. 19, it is clear that the translation of v along 
u is laterally a motion to the right side, while that of u along 
V is to the left, the translated magnitude in both cases being 
that with reference to which I speak of right and left. Thus 
the meaning of the equation v .u— — u.v is obvious. 
u 
-^v 
•'(S' 
III. Measurement and Summation of Translations. 
(35.) Units of Translation . — I shall take the translation of a pig. 20. 
unit along a perpendicular unit to be a unit of lateral translation ; 
and the translation of a unit along itself to be CL unit of longitudinal 
translation. Thus (see art. 16) a. (3, (3.y, a'. (3', &c. are units of 
lateral translation ; and aXa, (3 X (3, aJxod, &c. are units of longi- 
tudinal translation. 
