Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\vec{u}$ and $\vec{v}$ be unit vectors. if $\vec{w}$ is a vector such that $\vec{w} + (\vec{w} \times \vec{u}) = \vec{v}$, if the maximum volume of the parallelepiped formed by $\vec{u}, \vec{v}$ and $\vec{w}$ is $p$ then $12p = _______.
Step-by-Step Solution
Key Concept: Use dot and cross product operations to extract scalar information from vector equations; maximize expressions by analyzing constraints.
Given $\vec{w} + (\vec{w} \times \vec{u}) = \vec{v}$, taking dot product with $\vec{v}$ and $\vec{w}$ yields $|\vec{u} \vec{v}\vec{w}| = 1 - |\vec{w}|^2$ and $|\vec{w}|^2 = |\vec{v} \cdot \vec{w}|$. From $|\vec{w} \times \vec{u}|^2 = |\vec{u} - \vec{w}|^2$, we get $|\vec{w}|^2 = \frac{1}{1 + \sin^2 \theta}$, yielding $\max|\vec{u} \vec{v}\vec{w}| = \frac{1}{2}$.
Correct Answer: 6