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Question #67473

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be mutually orthogonal vectors in $\mathbb{R}^3$ with magnitudes $1, 2,$ and $3$ respectively. A new set of vectors $\vec{u}, \vec{v}, \vec{w}$ is defined as: $\vec{u} = 2\vec{a} + \vec{b}$ $\vec{v} = \vec{b} + 2\vec{c}$ $\vec{w} = 2\vec{a} + \vec{c}$ Let $V_1$ be the volume of the parallelepiped formed by $\vec{u}, \vec{v}, \vec{w}$ . Let $V_2$ be the volume of the parallelepiped formed by the unit vectors $\hat{u}, \hat{v}, \hat{w}$ which are in the directions of $\vec{u}, \vec{v}, \vec{w}$ respectively. Find the value of $V_1 + 260 V_2^2$ .

Options

A.
117
B.
81
C.
126
D.
144

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

Elementary Row Operations|Scalar Triple Product Volume Logic|Unit Vector Normalization

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the understanding of scalar triple products under linear transformations and the 'Unit Vector Normalization' trap.

Students often extract the coefficient matrix $M$ and treat its rows as the actual vectors in a standard orthonormal basis.

Consequently, they normalize the rows of $M$ directly (e.g., dividing by $\sqrt{5}$ ) to find the volume of the unit vectors, completely ignoring that the basis vectors $\vec{a}, \vec{b}, \vec{c}$ have different magnitudes.

The correct approach requires calculating the true magnitudes of the transformed vectors using the given non-uniform magnitudes of the orthogonal basis before scaling the volume.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Elementary Row Operations
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free