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Question #40470

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \cos \theta \hat{i} + \sin \theta \hat{j} + \hat{k}$ and $\vec{b} = -\sin \theta \hat{i} + \cos \theta \hat{j} + \hat{k}$ , where $\theta = \frac{4\pi}{3}$ . Let $\hat{n} = n_1\hat{i} + n_2\hat{j} + n_3\hat{k}$ be the unit normal vector to the plane containing $\vec{a}$ and $\vec{b}$ such that $\hat{n}$ makes an obtuse angle with the positive $z$ -axis. If $Z = n_1 + i n_2$ is a complex number in the Argand plane, then the principal argument of $Z$ is:

Options

A.
$-\frac{5\pi}{12}$
B.
$\frac{7\pi}{12}$
C.
$\frac{5\pi}{12}$
D.
$-\frac{7\pi}{12}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Argument and Principal Value|Geometric Interpretation of Cross Product|Unit Vector Normalization

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of the cross product mapped to the Argand plane, specifically focusing on the normalization and orientation trap.

Students often compute the cross product but fail to account for the 'obtuse angle with the z-axis' condition, which requires multiplying the normal vector by -1.

This sign flip reflects the complex number across the origin ( $Z \to -Z$ ), shifting its argument by $\pi$ .

Furthermore, it tests the strict adherence to the principal branch cut $(-\pi, \pi]$ of the Argand plane, forcing a transition from $\frac{19\pi}{12}$ to $-\frac{5\pi}{12}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Argument and Principal Value
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free