Question

Let $\theta$ be the angle between the unit vectors $\hat{a}$ and $\hat{b}$. If $|\hat{a} - \hat{b}| = \frac{\sqrt{3}}{2}$, then the value of $\cos \theta$ is

• A. $\frac{3}{8}$
• B. $\frac{1}{2}$
• C. $\frac{5}{8}$
• D. $\frac{3}{4}$
• E. $\frac{7}{8}$

Hint:

For any unit vectors $\hat{a}, \hat{b}$, the identity \( |\hat{a} - \hat{b}| = 2 \sin(\theta/2) \) is very useful. Here \( 2 \sin(\theta/2) = \sqrt{3}/2 \implies \sin(\theta/2) = \sqrt{3}/4 \). Then use \( \cos \theta = 1 - 2 \sin^2(\theta/2) \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We relate the magnitude of the difference of two vectors to the angle between them using the dot product expansion.
Step 2: Key Formula or Approach:
Square the given magnitude equation: \(|\hat{a} - \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 - 2(\hat{a} \cdot \hat{b})\).
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, \(|\hat{a}| = |\hat{b}| = 1\).
Also, \(\hat{a} \cdot \hat{b} = |\hat{a}||\hat{b}|\cos\theta = \cos\theta\).
Step 3: Detailed Explanation:
Given \(|\hat{a} - \hat{b}| = \frac{\sqrt{3}}{2}\).
Square both sides:
\[ |\hat{a} - \hat{b}|^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \] Expand the left side:
\[ |\hat{a}|^2 + |\hat{b}|^2 - 2(\hat{a} \cdot \hat{b}) = \frac{3}{4} \] Substitute the magnitudes of the unit vectors:
\[ 1 + 1 - 2\cos\theta = \frac{3}{4} \] \[ 2 - 2\cos\theta = \frac{3}{4} \] Rearrange to solve for \(\cos\theta\):
\[ 2\cos\theta = 2 - \frac{3}{4} \] \[ 2\cos\theta = \frac{8}{4} - \frac{3}{4} = \frac{5}{4} \] Divide by 2:
\[ \cos\theta = \frac{5}{8} \] Step 4: Final Answer:
The value of \(\cos\theta\) is \(\frac{5}{8}\).