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r r 2 r r 2 r 2 r 2
uv+ + uv− = 2 u + 2 v

Pb. 7.6 Referring to the inequality of Eq. (7.43), which relates the equivalent
resistances of n-resistors in series and in parallel, under what conditions does
the equality hold?

7.5 Cross Product and Scalar Triple Product*

In this section and in Sections 7.6 and 7.7, we restrict our discussions to vectors
in a 3-D space, and use the more familiar conventional vector notation.

7.5.1 Cross Product
r r
DEFINITION If two vectors are given by u = ( u u u ) and , 2 , 3 v = ( v v v, 2 , 3 )
r
r
1
1
then their cross product, denoted by uv× , is a vector given by:
r r
u v×= ( uv − uv uv −, u v u v −, uv ) (7.49)
23 3 2 3 1 1 3 1 2 2 1
By simple substitution, we can infer the following properties for the cross
product as summarized in the preparatory exercises below.

Preparatory Exercises

Pb. 7.7 Show, using the above definition for the cross product, that:
rr r r r r r r r r
a. u u v⋅( × ) = v uv⋅( × ) = ⇒0 uv× is orthogonal to both and v
u
r r 2 r 2 r 2 r r
b. uv× = u v − ( u v⋅ ) 2 Called the Lagrange Identity
r r r r
c. uv×= −( v u× ) Noncommutativity
r r r r r r r
d. u × ( v + w =) u v u w× + × Distributive property
r r r r r r
e. ku v( × ) = ( ku) × = u (× kv)
v
r r r
f. u ×=0 0
r r r
g. uu× = 0
Pb. 7.8 Verify the following relations for the basis unit vectors:

© 2001 by CRC Press LLC