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Question #40859

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The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that $|\vec{a}| = |\vec{b}| = |\vec{c}| = 2$ and the angle between any two of them is $\frac{\pi}{3}$ . Let $\vec{a}^*, \vec{b}^*, \vec{c}^*$ be the corresponding reciprocal system of vectors. A tetrahedron has vertices at the origin $O$ and at points $P, Q, R$ with position vectors $\vec{p} = \vec{a}^* + 2\vec{b}^* + 3\vec{c}^*$ , $\vec{q} = 2\vec{a}^* + 3\vec{b}^* + \vec{c}^*$ , and $\vec{r} = 3\vec{a}^* + \vec{b}^* + 2\vec{c}^*$ . If $G$ is the centroid of the tetrahedron $OPQR$ , find the value of $32 |\vec{OG}|^2$ .

Options

A.
27
B.
48
C.
54
D.
216

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

Centroid of a Tetrahedron|Reciprocal System of Vectors|Parametric form pitfalls|Projection vs. Coordinate

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate basis transformations and avoid parametric form pitfalls in a reciprocal system.

A common trap is assuming the reciprocal vectors $\vec{a}^*, \vec{b}^*, \vec{c}^*$ behave like an orthonormal Cartesian basis, which incorrectly yields $|\vec{a}^* + \vec{b}^* + \vec{c}^*|^2 = 3$ and an answer of 216.

Another trap is assuming $|\vec{a}^*| = 1/|\vec{a}|$ while maintaining orthogonality, yielding 54.

The problem forces the explicit use of the metric tensor or a symmetry-based basis transformation to find the true norm in the reciprocal space, distinguishing scalar coordinate components from actual geometric lengths.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Centroid of a Tetrahedron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free