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Question #76203

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ be any four vectors in $\mathbb{R}^3$ . Consider the determinant $\Delta = \begin{vmatrix} \vec{a}\cdot\vec{c} & \vec{a}\cdot\vec{d} & [\vec{a}, \vec{c}, \vec{d}] \\ \vec{b}\cdot\vec{c} & \vec{b}\cdot\vec{d} & [\vec{b}, \vec{c}, \vec{d}] \\ [\vec{c}, \vec{b}, \vec{a}] & [\vec{d}, \vec{b}, \vec{a}] & (\vec{b}\times\vec{a})\cdot(\vec{c}\times\vec{d}) \end{vmatrix}$ where $[\vec{x}, \vec{y}, \vec{z}]$ denotes the scalar triple product. Which of the following is identically equal to $\Delta$ ?

Options

A.
$|\vec{a}\times\vec{b}|^2 |\vec{c}\times\vec{d}|^2$
B.
$-|\vec{a}\times\vec{b}|^2 |\vec{c}\times\vec{d}|^2$
C.
$((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}))^2$
D.
$- ((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}))^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Vector Product Anti-Commutativity|Determinant Product Rule

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's mastery of the Determinant Product Rule (Gram determinant extension) and the anti-commutativity of the vector cross product.

The core trap lies in the evaluation of the scalar triple product $[\vec{a}, \vec{b}, \vec{b}\times\vec{a}]$ .

Students frequently memorize the identity $[\vec{x}, \vec{y}, \vec{x}\times\vec{y}] = |\vec{x}\times\vec{y}|^2$ and hastily apply it without noticing the reversed order of vectors in the cross product.

By expressing the determinant as the product of two matrices, the row-swapping property forces a negative sign, yielding $-|\vec{a}\times\vec{b}|^2$ .

Failing to account for this sign inversion leads directly to the incorrect positive magnitude squared option.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Vector Product Anti-Commutativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free