Question

Let \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \) such that \( \sin \theta = \frac{3}{5} \). Then, \( \hat{a} \cdot \hat{b} \) is equal to:

• A. \( \pm \frac{3}{5} \)
• B. \( \pm \frac{3}{4} \)
• C. \( \pm \frac{4}{5} \)
• D. \( \pm \frac{4}{3} \)

Hint:

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find one trigonometric value from another.

Answer 1

The Correct Option is C

Step 1: Use trigonometric identity. We know: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Step 2: Substitute values and solve. Given \( \sin \theta = \frac{3}{5} \), we get: \[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{9}{25} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = \frac{16}{25} \] \[ \cos \theta = \pm \frac{4}{5} \] Step 3: Apply the dot product formula. For unit vectors: \[ \hat{a} \cdot \hat{b} = \cos \theta = \pm \frac{4}{5} \] Final Answer: \[ \boxed{\pm \frac{4}{5}} \]