Question

If \( |\vec{a} + \vec{b}| = \frac{\sqrt{14}}{2} \) where \( \vec{a} \) and \( \vec{b} \) are unit vectors, then the value of \( |\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 \) is equal to

• A. \(3 \)
• B. \(4 \)
• C. \( \sqrt{5} \)
• D. \( \sqrt{7} \)
• E. \(7 \)

Hint:

Use identities for \( |\vec{a}\pm\vec{b}|^2 \) to avoid expanding vectors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves vector identities related to the magnitudes of sums and differences of vectors. Specifically, it uses the identity that relates \(|\vec{u} + \vec{v}|^2 - |\vec{u} - \vec{v}|^2\) to the dot product \(\vec{u} \cdot \vec{v}\).
Step 2: Key Formula or Approach:
1. Expand \(|\vec{a} + \vec{b}|^2\) as \((\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b})\). 2. Expand \(|\vec{a} - \vec{b}|^2\) as \((\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b})\). 3. The expression simplifies to \(|\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 = 4(\vec{a} \cdot \vec{b})\). 4. Use the given information to find the value of \(\vec{a} \cdot \vec{b}\).
Step 3: Detailed Explanation:
1. Use the vector identity. The expression we need to evaluate is: \[ |\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 \] This simplifies to: \[ (|\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b})) - (|\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b})) = 4(\vec{a} \cdot \vec{b}) \] So, our goal is to find the value of \(4(\vec{a} \cdot \vec{b})\). 2. Find the value of \(\vec{a} \cdot \vec{b}\). We are given that \(\vec{a}\) and \(\vec{b}\) are unit vectors, which means \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\). We are also given \(|\vec{a} + \vec{b}| = \frac{\sqrt{14}}{2}\). Let's square this: \[ |\vec{a} + \vec{b}|^2 = \left(\frac{\sqrt{14}}{2}\right)^2 = \frac{14}{4} = \frac{7}{2} \] Now use the expansion for \(|\vec{a} + \vec{b}|^2\): \[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \] Substitute the known values: \[ \frac{7}{2} = 1^2 + 1^2 + 2(\vec{a} \cdot \vec{b}) \] \[ \frac{7}{2} = 2 + 2(\vec{a} \cdot \vec{b}) \] Solve for \(2(\vec{a} \cdot \vec{b})\): \[ 2(\vec{a} \cdot \vec{b}) = \frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2} \] This gives us the value of \(\vec{a} \cdot \vec{b} = \frac{3}{4}\). 3. Calculate the final expression. The expression we need is \(4(\vec{a} \cdot \vec{b})\). \[ 4(\vec{a} \cdot \vec{b}) = 4 \left(\frac{3}{4}\right) = 3 \] Step 4: Final Answer:
The value of the expression is 3.