Question

If \(\hat{u}\) and \(\hat{v}\) are two non-collinear unit vectors such that \(\left| \frac{\hat{u} + \hat{v}}{2} + \hat{u} \times \hat{v} \right| = 1\), then the value of \(\left| \hat{u} \times \hat{v} \right|\) is equal to:

• A. \(\left| \frac{\hat{u} + \hat{v}}{2} \right|\)
• B. \(\left| \hat{u} + \hat{v} \right|\)
• C. \(\left| \hat{u} - \hat{v} \right|\)
• D. \(\left| \frac{\hat{u} - \hat{v}}{2} \right|\)

Hint:

When dealing with unit vectors and their cross product, remember that the magnitude of the cross product equals the sine of the angle between them, providing the area of the parallelogram they span.

Answer 1

The Correct Option is D

Given that, \[ \left| \frac{\hat{u} + \hat{v}}{2} + \hat{u} \times \hat{v} \right| = 1 \] we have: \[ \left| \frac{\hat{u} + \hat{v}}{2} \right|^2 + \left| \hat{u} \times \hat{v} \right|^2 = 1 \] Expanding and simplifying, \[ \frac{2 + 2\cos \theta}{4} + \sin^2 \theta = 1 \quad {since } \hat{u} \cdot \hat{v} = \cos \theta { and } \left| \hat{u} \times \hat{v} \right| = \sin \theta \] This implies: \[ \cos^2 \frac{\theta}{2} = \cos \theta \] Hence, \[ \theta = n\pi \pm \frac{\theta}{2}, \quad n \in \mathbb{Z} \] For \(n=1\), we find \(\theta = \frac{2\pi}{3}\). Thus, \[ \left| \hat{u} \times \hat{v} \right| = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \] This equals: \[ \left| \frac{\hat{u} - \hat{v}}{2} \right| \]