Page 203 -

r
0 0 =)
0 = (, 0ê + 0ê (7.2)
1 2
r r r
uv+ = w = ( u + v u +, v ) = ( u + v ê +) ( u + v ê) (7.3)
1 1 2 2 1 1 1 2 2 2
r
ku = ( ku ku =, ) ( ku ê +) ( ku ê) (7.4)
1 2 1 1 2 2

Preparatory Exercise

Pb. 7.1 Using the above definitions and properties, prove the following
identities:

r r r r
+
+
uv = v u
r r r r r r
+
( uv +) w = u + ( v + w)
r r r r r
u +=+ u = u
0
0
r r r
u +−( u =) 0
r r
klu =() ( kl u )
r r r r
+
ku v =( + ) ku kv
r r r
+
+
( kl u =) ku lu
The norm of a vector is the length of this vector. Using the Pythagorean the-
orem, its square is:

r 2
u = u + u 2 (7.5)
2
1 2
r
and therefore the unit vector in the u direction, denoted by ê , is given by:
u
1
ê = (, (7.6)
uu )
u 1 2
u + u 2
2
1 2
All of the above can be generalized to 3-D, or for that matter to n-dimensions.
For example:
1
ê = (, u , ) (7.7)
uu …,
u 1 2 n
u + u +… u 2
2
2
1 2 n
© 2001 by CRC Press LLC