Moment of inertia & Product of inertia:

The angular momentum of \( \vec{L} \) of a rigid body in vector notation can be represented as

\( \vec{L}=\overleftrightarrow{I}\vec{\omega} \)

where, \( \overleftrightarrow{I}=\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I{zy}&I_{zz}\end{bmatrix} \) is the moment of inertia tensor.

The three diagonal elements of the moment of inertia tensor \( I_{xx} \), \( I_{yy} \) and \( I_{zz} \) are called the moment of inertia of the rigid body about the rigid body about X, Y and Z axes respectively.

On the other hand the six off-diagonal elements \( I_{xy} \), \( I_{xz} \), \( I_{yx} \), \( I_{yz} \), \( I_{zx} \) and \( I_{zy} \) are called the product of inertia of the rigid body. This occurs in the symmetric pairs i.e.,

\( I_{xy}=I_{yx} \), \( I_{yz}=I_{zy} \) and \( I_{xz}=I_{zx} \).