5.4 (by hand)
5.4 The moment of inertia $I_x$, $I_y$, and the product of inertia $I_{xy}$ of the
cross-sectional area shown in the figure are:
$I_x = 5286 \text{ mm}^4$, $I_y = 4331 \text{ mm}^4$, and $I_{xy} = 2914 \text{ mm}^4$
The principal moments of inertia are the eigenvalues of the matrix
$\begin{bmatrix} 5286 & 2914 \ 2914 & 4331 \end{bmatrix}$, and the principal axes are in the direction of the eigen-
vectors. Determine the principal moments of inertia by solving the char-
acteristic equation. Determine the orientation of the principal axes of
inertia (unit vectors in the directions of the eigenvectors).