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Question #40297

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$ such that the dot product of any two of them is non-zero. A set $S$ is defined as $S = \{\vec{a}, \vec{b}, \vec{c}, \vec{p}, \vec{q}, \vec{r}\}$ , where $\vec{p} = \vec{a} \times (\vec{b} \times \vec{c})$ , $\vec{q} = \vec{b} \times (\vec{c} \times \vec{a})$ , and $\vec{r} = \vec{c} \times (\vec{a} \times \vec{b})$ . What is the number of proper subsets of $S$ that span $\mathbb{R}^3$ ?

Options

A.
37
B.
38
C.
40
D.
41

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Subset vs. Proper Subset|Vector Triple Product Expansion|Linear Independence vs Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of set theory, linear algebra, and vector calculus.

The primary trap lies in counting the non-spanning subsets.

Students often use the Principle of Inclusion-Exclusion on the 4 coplanar planes, yielding 23 non-spanning subsets and $64 - 23 = 41$ spanning subsets.

This critically misses that subsets of size 2 (like $\{\vec{a}, \vec{p}\}$ ) are not in any single coplanar plane but still fail to span $\mathbb{R}^3$ .

Furthermore, students frequently forget to subtract 1 for the 'proper' subset condition, leading to incorrect answers like 38, 40, or 41.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Subset vs. Proper Subset
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free