AIIMS 2019 Physics Vector Algebra MCQ Question

Option D is correct because if $\cos\theta = 0$, then $\theta = 90^\circ$, indicating that vectors $\vec{A}$ and $\vec{B}$ are perpendicular. This results in $| \vec{A} + \vec{B} |^2 = | \vec{A} - \vec{B} |^2$, as both expressions simplify to $| \vec{A} |^2 + | \vec{B} |^2$.

Option A is incorrect because it does not hold true for all vectors unless $\cos\theta = 0$. Option B is also incorrect, as it suggests that the two expressions are equal without considering the implications of the cosine term. Option C, stating $4AB \cos\theta = 0$, is true but does not directly lead to the conclusion about the angle $\theta$ being $90^\circ$.