Question

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

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

Hint:

Remember that \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this identity to compute \( \cos \theta \) when \( \sin \theta \) is given.

Answer 1

The Correct Option is C

The dot product of two unit vectors is given by: \[ \mathbf{\hat{a}} \cdot \mathbf{\hat{b}} = \cos \theta. \] 

Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1. \] 

Substitute \( \sin \theta = \frac{3}{5} \): \[ \left( \frac{3}{5} \right)^2 + \cos^2 \theta = 1. \] Simplify: \[ \frac{9}{25} + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}. \] 

Thus: \[ \cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}. \] 

Final Answer: \( \mathbf{\hat{a}} \cdot \mathbf{\hat{b}} = \pm \frac{4}{5} \), (C).