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vt() = v 0( )
1 1
β
vt( ) = v 0( )cos(α t) + v 0( )sin(α t) + 1 [ − cos(α t)]
2 2 3 α
β
vt( ) =− v 0( )sin(α t) + v 0( )cos(α t) + sin(α t)
3 2 3 α

Homework Problems

Pb. 8.23 Plot the 3-D curve, with time as parameter, for the tip of the veloc-
r v
ity vector of an electron with an initial velocity v()0 = 0 ê ( 1 + ê + ê ), where
2
3
3
v = 10 m/s, entering a region of space where the electric field and the mag-
5
0
r
netic flux density are constant and described by E = E ê , where E = –10 4
0
0 3
r
–2
2
V/m; and B = B ê , where B = 10 Wb/m .
0
0 1
Pb. 8.24 Find the parametric equations for the position vector for Pb. 8.23,
assuming that the origin of the axis is where the electron enters the region of
the force fields. Plot the 3-D curve that describes the position of the electron.
8.9.4 Pauli Spinors
We have shown thus far in this section the power of the Cayley-Hamilton the-
orem in helping us avoid the explicit computation of the eigenvectors while
still analytically solving a number of problems of linear algebra where the
dimension of the matrices was essentially 2 ⊗ 2, or in some special cases 3 ⊗
3. In this subsection, we discuss another analytical technique for matrix
manipulation, one that is based on a generalized underlying abstract alge-
braic structure: the Pauli spin matrices. This is the prototype and precursor to
more advanced computational techniques from a field of mathematics called
Group Theory. The Pauli matrices are 2 ⊗ 2 matrices given by:

0  1
σ =   (8.41a)
1  1 0

0  − 1
σ = j   (8.41b)
2  1 0 

 1 0 
σ =   (8.41c)
3 0  − 1

© 2001 by CRC Press LLC