For any to vectors ` vec A and vec B`, prove that
|vec A xx vec B|^2 = A^2 B^2 - ( vec A. vec B)^2`.

Explanation

To prove the identity \( |\vec{A} \times \vec{B}|^2 = A^2 B^2 - (\vec{A} \cdot \vec{B})^2 \), we will start from the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step-by-Step Solution:

1. **Start with the LHS**:
\[ |\vec{A} \times \vec{B}|^2 \]
...