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Chapter 3: Problem 30

The image of the point \((3,2,-1)\) in the YOZ plane is

Short Answer

Expert verified
The image is (-3, 2, -1).

Step by step solution

01

Understand the YOZ Plane

The YOZ plane is a vertical plane in a 3-dimensional space where the x-coordinate is zero. In other words, any point on this plane has a form \((0, y, z)\).
02

Identify the Original Point

The original point given is \((3, 2, -1)\). Here, \( x = 3 \), \( y = 2 \), and \( z = -1 \).
03

Reflect the Point Across the YOZ Plane

To find the image of a point in the YOZ plane, we set the x-coordinate to its negative value while keeping the y and z coordinates the same. Thus, the reflected point will be \((-3, 2, -1)\).
04

Verify the Reflected Point

Check whether the new x-value corresponds to a reflection about the YOZ plane. Since the line of reflection changes the sign of the x-coordinate, our new point \((-3, 2, -1)\) is correct, as its x-coordinate is the negative of the original point's x-coordinate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

YOZ Plane
The YOZ plane is an integral part of 3D Geometry. It represents a vertical plane in the three-dimensional space where the x-coordinate is always zero. This means any point lying in this plane will have the format
  • a coordinate of zero on the x-axis,
  • and any values for the y and z coordinates.
For example, the points
  • y and
  • z are free to vary while x remains zero, indicating presence in the YOZ plane.
Understanding the YOZ plane is crucial for solving problems involving reflections and transformations in 3D Coordinate Geometry. It is a simple yet fundamental concept that helps you navigate through 3D spaces with ease.
Point Reflection
Point reflection in geometry involves flipping a point over a specific axis or plane. For example, reflecting a point across the YOZ plane affects only the x-coordinate. Given a point
  • \((x, y, z)\), the reflected point will have its x-coordinate negated.
  • So if your original point is \((3, 2, -1)\), the reflection in the YOZ plane is
  • \((-3, 2, -1)\).
This transformation keeps the y and z coordinates unchanged.
Understanding the reflection process helps in visualizing and solving geometry problems involving symmetry and transformations effectively.
Coordinate System
A coordinate system is the foundation of understanding geometry and spatial relationships. In a 3D coordinate system, every point in space is represented by three coordinates:
  • x, y, and z.
These coordinates define the position of a point in three-dimensional space where:
  • x represents the position along the horizontal axis,
  • y along the vertical axis,
  • and z represents depth.
The position of any given point can be defined uniquely in this system. For example, the point \((3, 2, -1)\) indicates that it is
  • 3 units in the positive x-direction,
  • 2 units in the positive y-direction, and
  • 1 unit in the negative z-direction.
Grasping this concept of a coordinate system makes analyzing problems on reflection, translation, and rotation simpler and more intuitive.
Mathematics Education
Mathematics education often involves teaching concepts that help students develop problem-solving skills. A focus on geometry, especially 3D concepts like point reflection and planes, develops spatial awareness. It helps students:
  • Understand complex spatial relationships,
  • and apply mathematical principles to real-world problems.
Providing clear explanations of geometric transformations and coordinate systems enables students to visualize mathematical ideas. This also aids in
  • enhancing logical thinking,
  • boosting creativity through understanding patterns, and
  • encouraging persistence in challenging academic endeavors.
Effective teaching strategies in mathematics education ensure that students do not just memorize rules but also understand the reasoning behind mathematical procedures.

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