Question

\(\mathrm{A}^{\rightarrow}\) 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) \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}=\mathrm{A}^{2}-\mathrm{B}^{2}\)

Short answer

None of the given options are correct. The actual simplification for the expression \(\left|\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right|^{2}-\left|\mathrm{A}^{\rightarrow}-\mathrm{B}^{\rightarrow}\right|^{2}\) is \(4\mathrm{A}\cdot\mathrm{B}\).

Step-by-step solution

Option A

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

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

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

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 \(4\mathrm{A}\cdot\mathrm{B}\).

Key concepts

Vector Addition

Vector addition is a fundamental concept in vector mathematics that allows us to combine vectors to form a resultant vector. When we add two vectors, we essentially join them tip-to-tail without changing their magnitudes or directions. This operation results in a new vector that represents the cumulative effect of the original two vectors. For example, in physics, this could help determine the net force or displacement when multiple forces or movements act on an object.

The vector addition process can be visualized graphically or solved algebraically.

• Graphical Method: Position the tail of the second vector at the tip of the first vector. The vector sum is the vector from the tail of the first vector to the tip of the second vector.
• Algebraic Method: If vectors are expressed in components, their addition is straightforward. If \(\mathbf{A} = (A_x, A_y)\) and \(\mathbf{B} = (B_x, B_y)\), then their sum \(\mathbf{C} = \mathbf{A} + \mathbf{B}\) is given by \[\mathbf{C} = (A_x + B_x, A_y + B_y)\].

Understanding vector addition is crucial for solving problems involving multiple forces or movements, as it provides a straightforward method to determine their combined effect.

Vector Magnitude

The magnitude of a vector represents the length or size of the vector. It is always a positive value and provides an understanding of the "amount" or "intensity" of what the vector represents, be it force, velocity, or any other vector quantity.

Mathematically, the magnitude of a vector is determined by taking the square root of the sum of the squares of its components. For a vector \(\mathbf{A} = (A_x, A_y)\), its magnitude \(\left| \mathbf{A} \right|\) is given by \[\left| \mathbf{A} \right| = \sqrt{A_x^2 + A_y^2}\].

• 3D Vectors: For vectors in three dimensions, the magnitude formula extends to include the z-component, \[\left| \mathbf{A} \right| = \sqrt{A_x^2 + A_y^2 + A_z^2}\].

Understanding the magnitude is important for appropriately scaling and normalizing vectors, which is often required in applications like computer graphics and physics simulations.

Dot Product

The dot product is a scalar value derived from two vectors, which provides a way to assess the extent of their alignment. It is calculated by multiplying corresponding components of two vectors and summing these products. The mathematical expression for the dot product of two vectors \(\mathbf{A} = (A_x, A_y)\) and \(\mathbf{B} = (B_x, B_y)\) is given by:\[\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y\].

This results in a singular value, not another vector, which makes the dot product distinct compared to vector addition.

• Geometrical Interpretation: In terms of angle, the dot product can also be expressed as \[\mathbf{A} \cdot \mathbf{B} = \left|\mathbf{A}\right| \left|\mathbf{B}\right| \cos(\theta)\], where \(\theta\) is the angle between the two vectors.
• Applications: The dot product is particularly useful in determining the angle between vectors, assessing parallelism, and projecting one vector onto another.

An essential aspect of working with dot products is their ability to simplify expressions as shown in the original exercise. They help in calculating the squared magnitude difference through their involvement in expansions.