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Question #41895

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all non-zero vectors in $\mathbb{R}^3$ . A relation $R$ is defined on $S$ such that for any $\vec{u}, \vec{v} \in S$ , $\vec{u} R \vec{v}$ if and only if $\vec{u} \times (\vec{u} \times \vec{v}) + 2|\vec{u}|^2 \vec{v} = \vec{v} \times (\vec{v} \times \vec{u}) + 2|\vec{v}|^2 \vec{u}$ Which of the following statements is true regarding the relation $R$ ?

Options

A.
$R$ is symmetric but not antisymmetric
B.
$R$ is antisymmetric but not symmetric
C.
$R$ is both symmetric and antisymmetric
D.
$R$ is neither symmetric nor antisymmetric

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Symmetry vs. Antisymmetry|Linear Independence vs Dependence|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

Students observe the structural symmetry of the defining equation (swapping u and v yields the exact same equation) and immediately conclude the relation is symmetric.

When testing for antisymmetry, they often attempt to manipulate the dot products (projection conditions) without realizing that the equation forces linear dependence.

By missing the degenerate case where v = λu, they fail to see that the relation simplifies strictly to the equality relation u = v.

Equality is the trivial state that is simultaneously symmetric and antisymmetric, trapping students who assume these properties are mutually exclusive in non-trivial vector relations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Symmetry vs. Antisymmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free