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Question #41614

Difficulty: 8/10 | 🟢 Free

The Question

Let a⃗=sin⁡3θi^+cos⁡3θj^+sin⁡θk^\vec{a} = \sin 3\theta \hat{i} + \cos 3\theta \hat{j} + \sin \theta \hat{k} and b⃗=sin⁡θi^+cos⁡θj^+sin⁡3θk^\vec{b} = \sin \theta \hat{i} + \cos \theta \hat{j} + \sin 3\theta \hat{k} be two vectors forming adjacent sides of a triangle, where θ∈(0,π4)\theta \in \left(0, \frac{\pi}{4}\right). If the area of the triangle is 322\frac{\sqrt{3}}{2\sqrt{2}}, then a unit vector perpendicular to the plane of the triangle is:

Options

  • A.

    −i^−3j^+2k^22\frac{-\hat{i} - \sqrt{3}\hat{j} + 2\hat{k}}{2\sqrt{2}}

  • B.

    i^−3j^+2k^22\frac{\hat{i} - \sqrt{3}\hat{j} + 2\hat{k}}{2\sqrt{2}}

  • C.

    −3i^−j^+2k^22\frac{-\sqrt{3}\hat{i} - \hat{j} + 2\hat{k}}{2\sqrt{2}}

  • D.

    −i^+3j^+2k^22\frac{-\hat{i} + \sqrt{3}\hat{j} + 2\hat{k}}{2\sqrt{2}}

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Unit Vector Normalization|Geometric Interpretation of Cross Product|Double and Triple Angle Ratios

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to recognize and apply double and multiple angle trigonometric identities within the geometric context of a vector cross product.

The primary trap lies in the algebraic expansion of the cross product components.

If a student fails to simplify expressions like cos⁡3θsin⁡3θ−cos⁡θsin⁡θ\cos 3\theta \sin 3\theta - \cos \theta \sin \theta into cos⁡4θsin⁡2θ\cos 4\theta \sin 2\theta or sin⁡23θ−sin⁡2θ\sin^2 3\theta - \sin^2 \theta into sin⁡4θsin⁡2θ\sin 4\theta \sin 2\theta using sum-to-product and double angle formulas, they will be left with massive, unmanageable algebraic terms.

This prevents the extraction of the common factor sin⁡2θ\sin 2\theta, making the magnitude calculation and subsequent unit vector normalization nearly impossible, thereby inducing calculation fatigue and cascading errors.

Quick Facts About This Question

Question Information

SubjectMathematics
ChapterVector Algebra - Class 12
ConceptUnit Vector Normalization
Difficulty8
Cognitive Load7
Target Solve Time160 sec
AccessFree