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. \( \pm \frac{4}{3} \)
• E.

Hint:

The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) is a fundamental trigonometric principle used to find \( \cos \theta \) when \( \sin \theta \) is known.

Answer 1

The Correct Option is C

The dot product of two unit vectors \( \hat{a} \) and \( \hat{b} \) is \( \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos \theta \). Since \( |\hat{a}| = |\hat{b}| = 1 \), this simplifies to \( \hat{a} \cdot \hat{b} = \cos \theta \). Given \( \sin \theta = \frac{3}{5} \) and using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we have \( (\frac{3}{5})^2 + \cos^2 \theta = 1 \), which means \( \frac{9}{25} + \cos^2 \theta = 1 \). Solving for \( \cos^2 \theta \), we get \( \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \). Therefore, \( \cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \). Consequently, the value of \( \hat{a} \cdot \hat{b} \) is \( \pm \frac{4}{5} \). The correct answer is (C) \( \pm \frac{4}{5} \).