A

Step 1: According to the given information, we are analyzing the conditions under which the equation \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \) holds true.

Step 2: This equation can be interpreted geometrically. The magnitudes of the resultant vector when two vectors are added or subtracted can only be equal when the vectors are either parallel or anti-parallel to each other.

Step 3: If \( \vec{A} \) and \( \vec{B} \) are parallel or anti-parallel, their directions will either be the same or diametrically opposed, thus making the resultant magnitudes equal under the stated condition.

Therefore, the correct answer is A: \( |\vec{A}| = |\vec{B}| \neq 0 \) (both vectors are non-zero and of equal magnitude, allowing for the equality to hold when they are in the same or opposite directions).