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Question #41224

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}$ and $\vec{v}$ be two unit vectors in $\mathbb{R}^3$ such that the angle between them is $\frac{2\pi}{3}$ . Let $L_1$ and $L_2$ be the lines passing through the origin parallel to $\vec{u}$ and $\vec{v}$ respectively. Let $\vec{w}$ be a unit vector such that the angle between the line $L_3$ (parallel to $\vec{w}$ ) and $L_1$ is $\frac{\pi}{3}$ , and the angle between $L_3$ and $L_2$ is also $\frac{\pi}{3}$ . If the algebraic projection of $\vec{w}$ on $\vec{u}$ is positive and the algebraic projection of $\vec{w}$ on $\vec{v}$ is negative, let $M$ be the $3 \times 3$ matrix whose rows are the components of $\vec{u}, \vec{v},$ and $\vec{w}$ in a standard orthonormal basis. The value of $|\det(M)|$ is:

Options

A.
$0$
B.
$\frac{1}{2}$
C.
$\frac{1}{\sqrt{2}}$
D.
$\frac{\sqrt{3}}{2}$

Check Your Answer

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

Singular vs Non-Singular Matrices|Dot Product Projection Sign|Angle between two lines

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the angle between two lines (always acute or right) and the angle between two directed vectors (can be obtuse).

Students often incorrectly assume that an angle of $\pi/3$ between lines implies a dot product of $+1/2$ for the direction vectors.

The trap explicitly forces them to use the sign of the algebraic projection to determine the correct dot products ( $\vec{w} \cdot \vec{u} = 1/2$ and $\vec{w} \cdot \vec{v} = -1/2$ ).

If a student ignores the projection sign and assumes both dot products are $1/2$ , the Gram determinant evaluates to $0$ , leading them to falsely conclude the matrix is singular (Option A).

This elegantly links vector geometry, projection signs, and linear dependence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Singular vs Non-Singular Matrices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free