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Question #41924

Difficulty: 8/10 | 🟢 Free

The Question

Consider two lines $L_1$ and $L_2$ in $\mathbb{R}^3$ having direction vectors $\vec{d_1} = (\sin \alpha + \sqrt{3}\cos \alpha)\hat{i} + (\cos \alpha - \sqrt{3}\sin \alpha)\hat{j} + 2\hat{k}$ and $\vec{d_2} = (\sin \beta)\hat{i} + (\cos \beta)\hat{j} - \hat{k}$ respectively, for some $\alpha, \beta \in \mathbb{R}$ . If the lines $L_1$ and $L_2$ are parallel, then the scalar projection of $\vec{d_1}$ on $\vec{d_2}$ is:

Options

A.
$-2\sqrt{2}$
B.
$2\sqrt{2}$
C.
$-\sqrt{2}$
D.
$\sqrt{2}$

Check Your Answer

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

Linear Independence vs Dependence|Angle between two lines|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the relationship between parallel lines and their direction vectors.

A common trap in JEE Advanced is assuming that parallel lines always have direction vectors with a positive scalar multiple relationship (i.e., the angle between them is 0).

Here, the $\hat{k}$ -components strictly force the scalar multiple to be negative ( $\lambda = -2$ ), making the vectors anti-parallel.

Students who merely calculate the magnitudes $|\vec{d_1}|$ and $|\vec{d_2}|$ and assume the scalar projection is positive because the lines are 'parallel' will incorrectly choose $2\sqrt{2}$ .

The question elegantly enforces the sign convention of the dot product and scalar projection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Linear Independence vs Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free