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Question #40304

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$ . Let $S = \{\vec{v}_1, \vec{v}_2, \vec{v}_3, \vec{v}_4, \vec{v}_5\}$ be a set of 5 distinct vectors in $\mathbb{R}^3$ such that $|\vec{v}_i| = r > 0$ for all $i=1, \dots, 5$ . Define the sets: $A = \{\vec{v} \in S \mid [\vec{v}, \vec{a}, \vec{b}] = 0\}$ $B = \{\vec{v} \in S \mid [\vec{v}, \vec{b}, \vec{c}] = 0\}$ $C = \{\vec{v} \in S \mid [\vec{v}, \vec{c}, \vec{a}] = 0\}$ Suppose $|A| = 4$ , $|B| = 3$ , and $|C| = 3$ . If the maximum absolute value of the scalar triple product of any three vectors in $S$ is 9, find the value of $\sum_{1 \le i < j < k \le 5} [\vec{v}_i, \vec{v}_j, \vec{v}_k]^2$

Options

A.
162
B.
243
C.
324
D.
810

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

Venn Diagram Overlap Logic|Linear Independence vs Dependence|Geometric Interpretation of Cross Product

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of set-theoretic Inclusion-Exclusion logic and linear algebra subspace constraints.

The primary trap lies in the evaluation of the sum of squares of the scalar triple products.

Students often assume the vectors are in general position and attempt to use Gram matrix determinants or assume all $\binom{5}{3}=10$ triplets contribute to the sum, leading to incorrect multiples like 810.

By applying Venn diagram overlap logic, the student must deduce that the coplanarity conditions force a specific collinearity distribution (2 vectors collinear with $\vec{a}$ , 2 with $\vec{b}$ , 1 with $\vec{c}$ ).

This geometric collapse forces 6 of the 10 triplets to have a volume of zero, leaving exactly 4 symmetric non-zero triplets that anchor the final calculation to $4 \times 9^2 = 324$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Venn Diagram Overlap Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free