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(σσ vw + σ σ v w ) = σ j ( v w− + v w ) (8.50)
1 3 1 3 3 1 3 1 2 1 3 3 1

(σσ vw + σσ vw ) = σ j (v w − v w ) (8.51)
2 3 2 3 3 2 3 2 1 2 3 3 2
r
Recalling that the cross product of two vectors (vw× r ) can be written from
Eq. (7.49) in components form as:

r r
(v w× ) = (vw − vw , v w− + vw , v w − vw )
2 3 3 2 1 3 3 1 1 2 2 1

the second, third, and fourth parentheses on the RHS of Eq. (8.47) can be com-
r
r
r
bined to give jσ⋅( v w× ), thus completing the proof of the theorem.
COROLLARY
If ê is a unit vector, then:

r
2
(σ⋅ ê ) = I (8.52)
PROOF Using Eq. (8.46), we have:

r r
2
(σ ⋅ ê ) = ( ⋅ê ê ) + jI σ ( ⋅ ê × ê ) = I
where, in the last step, we used the fact that the norm of a unit vector is one
and that the cross product of any vector with itself is zero.
A direct result of this corollary is that:
r
(σ⋅ ê ) 2 m = I (8.53)

and
r r
(σ ⋅ ê ) 2 m +1 = (σ ⋅ ê ) (8.54)

From the above results, we are led to the theorem:

THEOREM
r r
exp(jσ ⋅ êφ ) = cos( ) +φ jσ ⋅ ê sin( )φ (8.55)

PROOF If we Taylor expand the exponential function, we obtain:

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