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Question #40457

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \sqrt{2} \sin \theta \hat{i} + \sqrt{2} \cos \theta \hat{j} + \sqrt{3} \hat{k}$ and $\vec{b} = \sqrt{2} \cos \theta \hat{i} - \sqrt{2} \sin \theta \hat{j} + \sqrt{3} \hat{k}$ for some $\theta \in \mathbb{R}$ . A sequence of vectors is defined as $\vec{v}_1 = \vec{a} \times \vec{b}$ and $\vec{v}_k = \vec{a} \times \vec{v}_{k-1}$ for $k \ge 2$ . If $\vec{S} = \sum_{k=1}^{\infty} \left( \frac{1}{3} \right)^k \vec{v}_k$ , then the value of $49 |\vec{S}|^2$ is:

Options

A.
56
B.
196
C.
112
D.
144

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

Lagrange Identity Equivalents|Vector Triple Product Expansion|Scalar Multiplication Geometric Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle recursive vector triple products and geometric scaling.

The primary trap is the 'Scalar Multiplication Geometric Scaling' error, where students incorrectly assume $|\sum \vec{x}_k|^2 = \sum |\vec{x}_k|^2$ (leading to option B, 196).

A secondary trap involves ignoring the negative sign generated by the vector triple product expansion $\vec{a} \times (\vec{a} \times \vec{v}_1) = -|\vec{a}|^2 \vec{v}_1$ , which would lead to an incorrect geometric series sum.

The problem forces the use of Lagrange's identity to find the initial magnitude and relies on recognizing the mutually orthogonal components to compute the final magnitude squared.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Lagrange Identity Equivalents
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free