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Question #40764

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(2, 3, 4)$ be a point in space. Let $\Pi$ be the plane $x + y + z = 3$ and $L$ be the line $\frac{x-1}{1} = \frac{y-1}{-1} = \frac{z-1}{1}$ . If $P_1$ is the reflection of $P$ across the plane $\Pi$ , and $P_2$ is the reflection of $P_1$ across the line $L$ , then the coordinates of $P_2$ are:

Options

A.
$\left(\frac{16}{3}, \frac{5}{3}, \frac{10}{3}\right)$
B.
$\left(-\frac{2}{3}, -\frac{7}{3}, \frac{4}{3}\right)$
C.
$\left(\frac{8}{3}, \frac{13}{3}, \frac{2}{3}\right)$
D.
$(0, -9, -2)$

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

Reflection Principle|Dot Product Projection Sign|Collinearity vs Coplanarity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between reflection across a plane and reflection across a line in 3D space, specifically targeting the 'Collinearity vs Coplanarity' and 'Dot Product Projection Sign' traps.

Students often memorize the vector reflection formula $\vec{v}' = \vec{v} - 2(\vec{v} \cdot \hat{n})\hat{n}$ for planes and incorrectly apply it to lines, failing to realize that line reflection requires isolating the orthogonal component within the coplanar subspace, yielding $2(\vec{v} \cdot \hat{u})\hat{u} - \vec{v}$ .

This conceptual error leads directly to Option B.

Furthermore, when calculating the vector projection onto the line's direction, students frequently drop the negative sign of the dot product (scalar projection), resulting in a directional error that leads to Option A.

Option D catches those who incorrectly add instead of subtract the normal vector distance when reflecting across the initial plane.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Reflection Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free