Before You Begin

Question #41751

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ be the line of intersection of the planes $P_1: x + y + z = 1$ and $P_2: 2x - y + 3z = 4$ . Let $L_2$ be a line defined by the parametric equations $x = 1 + 2t$ , $y = -2 + t$ , $z = 3 - 2t$ , where $t \in \mathbb{R}$ . If $\theta$ is the acute angle between the lines $L_1$ and $L_2$ , then the value of $\sin^2 \theta$ is:

Options

A.
$\frac{5}{18}$
B.
$\frac{13}{18}$
C.
$\frac{1}{27}$
D.
$\frac{1}{14}$

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

Angle between two planes|Angle between two lines|Normal Form Parameters|Parametric Form Manipulation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify and compute the angle between two lines when one is given as the intersection of two planes and the other in parametric form.

The primary trap lies in the cognitive dissonance caused by the presence of planes: students frequently default to the 'angle between a line and a plane' formula (which uses $\sin\theta$ ) when planes are involved in the problem setup.

By asking for $\sin^2\theta$ , the problem baits students into directly squaring the dot product ratio (yielding the trap answer $\frac{13}{18}$ ), whereas the correct geometric relationship for two lines dictates using $\cos\theta$ for the dot product, making the true value $1 - \frac{13}{18} = \frac{5}{18}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Angle between two planes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free