Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\vec{u}$ and $\vec{v}$ are unit vectors and $\vec{w}$ is a vector such that $\vec{u} \times \vec{v} + \vec{u} = \vec{w}$ and $\vec{w} \times \vec{u} = \vec{v}$. Then the value of $[\vec{u}\vec{v}\vec{w}] = _______.
Step-by-Step Solution
Key Concept: The scalar triple product yields $|u||v|\sin\theta = 1$ when vectors are orthogonal.
Given $\vec{u} \times \vec{v} + \vec{u} = \vec{w}$ and $\vec{w} \times \vec{u} = \vec{v}$, we first establish that $(\vec{u} \cdot \vec{v})\vec{u} = 0$ implying $\vec{u} \cdot \vec{v} = 0$. Then using $[u \ v \ w] = u. (v \times w) = u. (v \times (\vec{u} \times \vec{v} + \vec{u})) = v^2u^2 = 1$.
Correct Answer: 1