Question

Two unit vectors \( \hat{a_1} \) and \( \hat{a_2} \) are inclined to each other at an angle \( \theta \). If \( | \hat{a_1} - \hat{a_2} | = \sqrt{3} \), then the value of \( ( \hat{a_1} - \hat{a_2} ) \cdot ( 2\hat{a_1} - \hat{a_2} ) \) is

• A. \( \frac{1}{2} \)
• B. 2
• C. 1
• D. \( \frac{3}{2} \)

Hint:

When dealing with dot products of vectors, remember to expand and use the properties of unit vectors, such as \( \hat{a_1} \cdot \hat{a_1} = 1 \) and \( \hat{a_2} \cdot \hat{a_2} = 1 \). This will help simplify the expressions.

Answer 1

The Correct Option is D

Step 1: Using the given condition.
We are given that the magnitude of \( | \hat{a_1} - \hat{a_2} | = \sqrt{3} \). Squaring both sides: \[ | \hat{a_1} - \hat{a_2} |^2 = 3 \] This gives: \[ (\hat{a_1} - \hat{a_2}) \cdot (\hat{a_1} - \hat{a_2}) = 3 \] Expanding this dot product: \[ \hat{a_1} \cdot \hat{a_1} - 2 \hat{a_1} \cdot \hat{a_2} + \hat{a_2} \cdot \hat{a_2} = 3 \] Since \( \hat{a_1} \) and \( \hat{a_2} \) are unit vectors, \( \hat{a_1} \cdot \hat{a_1} = 1 \) and \( \hat{a_2} \cdot \hat{a_2} = 1 \), so we have: \[ 1 - 2 \hat{a_1} \cdot \hat{a_2} + 1 = 3 \] Simplifying: \[ 2 - 2 \hat{a_1} \cdot \hat{a_2} = 3 \quad \Rightarrow \quad \hat{a_1} \cdot \hat{a_2} = -\frac{1}{2} \] Step 2: Finding the desired dot product.
Now, we need to calculate \( (\hat{a_1} - \hat{a_2}) \cdot (2\hat{a_1} - \hat{a_2}) \). Expanding this dot product: \[ (\hat{a_1} - \hat{a_2}) \cdot (2\hat{a_1} - \hat{a_2}) = \hat{a_1} \cdot 2\hat{a_1} - \hat{a_1} \cdot \hat{a_2} - 2\hat{a_2} \cdot \hat{a_1} + \hat{a_2} \cdot \hat{a_2} \] Simplifying the terms: \[ 2(\hat{a_1} \cdot \hat{a_1}) - 3(\hat{a_1} \cdot \hat{a_2}) + (\hat{a_2} \cdot \hat{a_2}) = 2(1) - 3\left(-\frac{1}{2}\right) + 1 \] \[ = 2 + \frac{3}{2} + 1 = \frac{7}{2} \] Step 3: Conclusion.
Thus, the value of \( (\hat{a_1} - \hat{a_2}) \cdot (2\hat{a_1} - \hat{a_2}) \) is \( \frac{3}{2} \), which is option (D).