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 given by: \[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos \theta, \] where \( |\hat{a}| = |\hat{b}| = 1 \) (since they are unit vectors). Therefore: \[ \hat{a} \cdot \hat{b} = \cos \theta. \] We are given \( \sin \theta = \frac{3}{5} \). Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1, \] we find: \[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1, \] \[ \frac{9}{25} + \cos^2 \theta = 1, \] \[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}. \] Thus: \[ \cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}. \] Therefore, the value of \( \hat{a} \cdot \hat{b} \) is: \[ \hat{a} \cdot \hat{b} = \pm \frac{4}{5}. \] Hence, the correct answer is (C) \( \pm \frac{4}{5} \).