\(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are nonzero vectors. Which from the followings is true ? (A) \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}=2\left(\mathrm{~A}^{2}+\mathrm{B}^{2}\right)\) (B) \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}=2\left(\mathrm{~A}^{2}+\mathrm{B}^{2}\right)\) (C) \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}=\mathrm{A}^{2}+\mathrm{B}^{2}\) (D) \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}=\mathrm{A}^{2}-\mathrm{B}^{2}\)

Option A is: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = 2\left(\mathrm{A}^2 + \mathrm{B}^2 \right)\) First, let's find the square of the magnitudes of \(\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow\) and \(\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow\): 1. \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 = (\mathrm{A} + \mathrm{B})\cdot(\mathrm{A} + \mathrm{B}) = \mathrm{A}^2 + 2\mathrm{A}\cdot\mathrm{B} + \mathrm{B}^2\) 2. \(\left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = (\mathrm{A} - \mathrm{B})\cdot(\mathrm{A} - \mathrm{B}) = \mathrm{A}^2 - 2\mathrm{A}\cdot\mathrm{B} + \mathrm{B}^2\) Now let's compute the difference between the two squared magnitudes: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = (\mathrm{A}^2 + 2\mathrm{A}\cdot\mathrm{B} + \mathrm{B}^2) - (\mathrm{A}^2 - 2\mathrm{A}\cdot\mathrm{B} + \mathrm{B}^2) = 4\mathrm{A}\cdot\mathrm{B}\) Option A simplifies to \(4\mathrm{A}\cdot\mathrm{B}\), and it is different from the given expression. So option A is incorrect.

Option B is the same as Option A: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = 2\left(\mathrm{A}^2 + \mathrm{B}^2 \right)\) Option B also simplifies to \(4\mathrm{A}\cdot\mathrm{B}\), which is different from the given expression. So option B is also incorrect.

Option C is: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = \mathrm{A}^2 + \mathrm{B}^2\) Using the calculation of the difference between the squared magnitudes that we found in Option A, the actual simplification is: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = 4\mathrm{A}\cdot\mathrm{B}\) Option C is also incorrect because it does not simplify to the correct expression.

Option D is: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = \mathrm{A}^2 - \mathrm{B}^2\) Using the calculation of the difference between the squared magnitudes that we found in Option A, the actual simplification is: \(\left|\mathrm{A}^\rightarrow + \mathrm{B}^\rightarrow \right|^2 - \left|\mathrm{A}^\rightarrow - \mathrm{B}^\rightarrow \right|^2 = 4\mathrm{A}\cdot\mathrm{B}\) Option D is also incorrect because it does not simplify to the correct expression. None of the options given are correct. The actual simplification of the given difference between the squared magnitudes is \(4\mathrm{A}\cdot\mathrm{B}\).