Question

If the angle between two unit vectors $ \vec{A} $ and $ \vec{B} $ is $ \theta $, then \[ |\vec{A} + \vec{B}| = \]

• A. \( 2 \cos \frac{\theta}{2} \)
• B. \( 2 \sin \frac{\theta}{2} \)
• C. \( 0 \)
• D. \( \cos \frac{\theta}{2} \)

Hint:

When dealing with vectors and their magnitudes, use trigonometric identities to simplify expressions involving the angle between the vectors.

Answer 1

The Correct Option is A

The magnitude of \( \mathbf{A} + \mathbf{B} \) is given by: \[ |\mathbf{A} + \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2 |\mathbf{A}| |\mathbf{B}| \cos \theta} \] Since both \( \mathbf{A} \) and \( \mathbf{B} \) are unit vectors, \( |\mathbf{A}| = |\mathbf{B}| = 1 \), so the formula simplifies to: \[ |\mathbf{A} + \mathbf{B}| = \sqrt{1 + 1 + 2 \cos \theta} = \sqrt{2(1 + \cos \theta)} \] Using the trigonometric identity \( 1 + \cos \theta = 2 \cos^2 \frac{\theta}{2} \), we get: \[ |\mathbf{A} + \mathbf{B}| = \sqrt{4 \cos^2 \frac{\theta}{2}} = 2 \cos \frac{\theta}{2} \] Thus, the correct answer is: \[ \boxed{2 \cos \frac{\theta}{2}} \]