Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade None
Question:
If $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{c} \times \vec{b}) = 0$ then which of the following is always true:
$\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are necessary coplanar
at least one of $\vec{a}$ or $\vec{c}$ must lie in plane of $\vec{b}$ and $\vec{c}$
at least one of $\vec{b}$ or $\vec{c}$ must lie in plane of $\vec{a}$ and $\vec{x}$
at least one of $\vec{a}$ or $\vec{b}$ must lie in plane of $\vec{c}$ and $\vec{x}$
Step-by-Step Solution
Key Concept: Scalar quadruple product identity relates cross and dot products to scalar triple products.
From $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 0$, using the scalar triple product identity $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})$, we get $[\vec{a}\vec{b}\vec{d}][\vec{c}\vec{b}] = 0$, implying $[\vec{a}\vec{b}\vec{c}] = 0$.
Correct Answer: 2