Mathematical Tools for Physics

7—Operators and Matrices 176

The rotated~e 1 has two components, so

Rα(~e 1 ) =~e 1 cosα+~e 2 sinα=R 11 ~e 1 +R 21 ~e 2

This determines the first column of the matrix of components,

R 11 = cosα, and R 21 = sinα

Similarly the effect on the other basis vector determines the second column:

Rα(~e 2 ) =~e 2 cosα−~e 1 sinα=R 12 ~e 1 +R 22 ~e 2

Check:Rα(~e 1 ).Rα(~e 2 ) = 0.

R 12 =−sinα, and R 22 = cosα

The component matrix is then
(
Rα

)

=

(

cosα −sinα

sinα cosα

)

(11)

Components of Inertia
The definition, Eq. ( 3 ), and the figure preceding it specify the inertia tensor as the function that relates the
angular momentum of a rigid body to its angular velocity.

~L=

∫

dm~r×

(

~ω×~r

)

=I(~ω) (12)

Use the vector identity,

A~×(B~×C~) =B~(A~.C~)−C~(A~.B~) (13)

then the integral is

L~=

∫

dm

[

~ω(~r.~r)−~r(~ω.~r)

]

=I(~ω)