Question

Given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).

• A. \( 7 \)
• B. \( 4 \)
• C. \( 14 \)
• D. \( 3 \)

Hint:

Use the properties of unit vectors and the standard identities for the magnitudes of vector sums and differences to simplify such problems.

Answer 1

The Correct Option is A

We are given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors. We need to find the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \).

1. Step 1: Use the identity for \( |\mathbf{a} + \mathbf{b}|^2 \): By the definition of the magnitude of a vector, we know that: \[ |\mathbf{a} + \mathbf{b}|^2 = \mathbf{a} \cdot \mathbf{a} + 2 \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} \] Since \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, \( \mathbf{a} \cdot \mathbf{a} = 1 \) and \( \mathbf{b} \cdot \mathbf{b} = 1 \), so: \[ |\mathbf{a} + \mathbf{b}|^2 = 1 + 2 \mathbf{a} \cdot \mathbf{b} + 1 = 2 + 2 \mathbf{a} \cdot \mathbf{b} \]

2. Step 2: Use the identity for \( |\mathbf{a} - \mathbf{b}|^2 \): Similarly, for \( |\mathbf{a} - \mathbf{b}|^2 \), we have: \[ |\mathbf{a} - \mathbf{b}|^2 = \mathbf{a} \cdot \mathbf{a} - 2 \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = 1 - 2 \mathbf{a} \cdot \mathbf{b} + 1 = 2 - 2 \mathbf{a} \cdot \mathbf{b} \]

3. Step 3: Subtract the two expressions: Now, we subtract \( |\mathbf{a} - \mathbf{b}|^2 \) from \( |\mathbf{a} + \mathbf{b}|^2 \): \[ |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 = (2 + 2 \mathbf{a} \cdot \mathbf{b}) - (2 - 2 \mathbf{a} \cdot \mathbf{b}) \] Simplifying this: \[ = 2 + 2 \mathbf{a} \cdot \mathbf{b} - 2 + 2 \mathbf{a} \cdot \mathbf{b} = 4 \mathbf{a} \cdot \mathbf{b} \]

4. Step 4: Use the given value of \( |\mathbf{a} + \mathbf{b}| \): We are given that \( |\mathbf{a} + \mathbf{b}| = \frac{\sqrt{14}}{2} \), so: \[ |\mathbf{a} + \mathbf{b}|^2 = \left( \frac{\sqrt{14}}{2} \right)^2 = \frac{14}{4} = \frac{7}{2} \] Using the equation \( |\mathbf{a} + \mathbf{b}|^2 = 2 + 2 \mathbf{a} \cdot \mathbf{b} \), we substitute this value: \[ \frac{7}{2} = 2 + 2 \mathbf{a} \cdot \mathbf{b} \] Solving for \( \mathbf{a} \cdot \mathbf{b} \): \[ \mathbf{a} \cdot \mathbf{b} = \frac{3}{4} \]
5. Step 5: Final answer: Now substitute \( \mathbf{a} \cdot \mathbf{b} = \frac{3}{4} \) into the expression for \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \): \[ 4 \mathbf{a} \cdot \mathbf{b} = 4 \times \frac{3}{4} = 3 \] Thus, the value of \( |\mathbf{a} + \mathbf{b}|^2 - |\mathbf{a} - \mathbf{b}|^2 \) is \( 7 \).