Introduction to Vectors and Tensors Vol 2 (Bowen 246). - Index of

298 Chap. 9 • EUCLIDEAN MANIFOLDS
x − y = x− z+ z−y (43.2)
and
x − y = v (43.3)
Theorem 43.1. In a Euclidean point space E
() i x− x = 0
( ii) x− y = −( y−x)
( iii) if x − y = x ' −y ', then x− x' = y−y'
(43.4)
Proof. For (i) take x = y = z in (43.2); then
x− x = x− x+ x−x
which implies x− x = 0. To obtain (ii) take y = x in (43.2) and use (i). For (iii) observe that
x − y ' = x − y+ y− y' = x− x' + x' −y
'
from (43.2). However, we are given x − y = x ' −y ' which implies (iii).
The equation
x − y = v
has the property that given any v and y , x is uniquely determined. For this reason it is customary
to write
x = y + v (43.5)