Show that `|veca|vecb+|vecb|veca` is perpendicular to `|veca|vecb-|vecb|veca` for any two non zero vectors `veca and vecb.

Explanation

To show that \(|\vec{a}| \vec{b} + |\vec{b}| \vec{a}\) is perpendicular to \(|\vec{a}| \vec{b} - |\vec{b}| \vec{a}\) for any two non-zero vectors \(\vec{a}\) and \(\vec{b}\), we need to prove that the dot product of these two vectors is zero.

### Step-by-Step Solution:

1. **Define the vectors**:
Let:
\[
\vec{u} = |\vec{a}| \vec{b} + |\vec{b}| \vec{a}
...