Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
If $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a}\vec{b})\vec{b} = (4 - 2\beta - \sin \alpha)\vec{b} + (\beta^2 - 1)\vec{c}$ and $\vec{c} \cdot \vec{c} = \vec{a} \cdot \vec{c}$ where $\vec{b}$ and $\vec{c}$ are non-collinear and $\alpha, \beta$ are scalars then $\beta = _______.
Step-by-Step Solution
Key Concept: Express vector equations in component form by comparing coefficients of linearly independent vectors to establish scalar equations.
From the given conditions $\vec{a} \cdot (\vec{b} \times \vec{c}) + (\vec{a} \cdot \vec{b})\vec{c} = (4 - 2\vec{B} - \sin \alpha)\vec{b} + (\vec{b}^2 - 1)\vec{c}$, we decompose to get $\vec{a} \cdot \vec{b} = 4 - 2\vec{B} - \sin \alpha$ and $\vec{a} \cdot \vec{b} = 1 - \vec{b}^2$. Equating yields $\sin \alpha = \vec{b}^2 - 2\vec{B} + 2$, so $\sin \alpha = (\vec{B} - 1)^2 + 1$, giving $\vec{B} = 1$ and $\alpha = (4n + 1)\frac{\pi}{2}$.
Correct Answer: 1