Question

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}\). If \(\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})\), then \(\lambda\) is equal to:

• A. \(\frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v})\)
• B. \(\frac{2}{3}(\vec{A} \cdot \hat{u}) - \frac{1}{3}(\vec{A} \cdot \hat{v})\)
• C. \(\frac{4}{3}(\vec{A} \cdot \hat{u}) + \frac{2}{3}(\vec{A} \cdot \hat{v})\)
• D. \((\vec{A} \cdot \hat{u}) - \frac{1}{2}(\vec{A} \cdot \hat{v})\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the properties of dot products and cross products for unit vectors. Specifically, take the dot product of the expression for \(\vec{A}\) with \(\hat{u}\) and \(\hat{v}\) to form a system of linear equations in \(\lambda\).

Step 2: Key Formula or Approach:
1. \(|\hat{u} \times \hat{v}| = \sin \theta\).
2. \(\hat{u} \cdot (\hat{u} \times \hat{v}) = 0\).
3. Solve for \(\lambda\) using \(\vec{A} \cdot \hat{u}\) and \(\vec{A} \cdot \hat{v}\).

Step 3: Detailed Explanation:
\(\sin \theta = \sqrt{3}/2 \implies \theta = 60^\circ \implies \hat{u} \cdot \hat{v} = 1/2\).
\(\vec{A} \cdot \hat{u} = \lambda(\hat{u} \cdot \hat{u}) + (\hat{v} \cdot \hat{u}) + 0 = \lambda + 1/2 \implies \lambda = \vec{A} \cdot \hat{u} - 1/2\) ...(i)
\(\vec{A} \cdot \hat{v} = \lambda(1/2) + 1 + 0 = \lambda/2 + 1 \implies \lambda/2 = \vec{A} \cdot \hat{v} - 1 \implies \lambda = 2(\vec{A} \cdot \hat{v}) - 2\) ...(ii)
Multiplying (i) by 4 and (ii) by 2, and solving the linear relationship results in \(\lambda = \frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v})\).

Step 4: Final Answer:
The matching option is (A).