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Question #73293

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{e}_1, \vec{e}_2, \vec{e}_3$ be the standard orthonormal basis of $\mathbb{R}^3$ . Let $A$ and $B$ be two $3 \times 3$ orthogonal matrices. For $i \in \{1, 2, 3\}$ , let $\vec{w}_i = AB\vec{e}_i + BA\vec{e}_i$ (which lies along the internal angle bisector of the vectors $AB\vec{e}_i$ and $BA\vec{e}_i$ ). If $S = \text{span}(\vec{w}_1, \vec{w}_2, \vec{w}_3)$ , which of the following sets represents all possible values for the dimension of $S$ ?

Options

A.
$\{3\}$
B.
$\{1, 3\}$
C.
$\{1, 2, 3\}$
D.
$\{0, 1, 2, 3\}$

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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

Angle Bisector Direction|Linear Independence vs Dependence|Matrix Multiplication Non-Commutativity

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 ability to analyze the linear independence of vectors generated by non-commutative matrix transformations.

The primary trap is the assumption of commutativity ( $AB = BA$ ), which incorrectly simplifies the transformation to $2AB$ , leading students to conclude that the vectors are always linearly independent and the dimension is strictly 3.

A secondary trap is assuming that since $A$ and $B$ are arbitrary, the rank of $AB+BA$ can take any value, ignoring the structural constraints of the commutator $B^T A^T B A \in SO(3)$ .

By recognizing that the eigenvalues of a 3D rotation matrix dictate the rank of $I+C$ , students must deduce that the dimension can only be 1 or 3, completely bypassing the intuitive but false intermediate rank of 2.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Angle Bisector Direction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free