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3D Volume Scaling Interactive Visualizer

Run the 3D Volume Scaling Visualizer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This 3D visualization extends the concept of signed area to signed volume in three dimensions.

Key Concepts:

  • The determinant of a 3×3 matrix equals the signed volume of the parallelepiped formed by its column vectors
  • \(|\det(A)|\) = volume scaling factor
  • \(\det(A) > 0\): orientation preserved (green)
  • \(\det(A) < 0\): orientation reversed (red)
  • \(\det(A) = 0\): volume collapses (singular)

How to Use

  1. Click preset buttons: Scaling, Rotation, or Singular matrix
  2. Drag to rotate: Click and drag in the 3D view to rotate camera
  3. Use morph slider: Animate from identity to target matrix
  4. Toggle unit cube: Show/hide the reference unit cube

Embedding

1
<iframe src="https://dmccreary.github.io/linear-algebra/sims/volume-scaling-3d/main.html" height="482px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Visualize the determinant as a volume scaling factor in 3D
  2. Distinguish between orientation-preserving and orientation-reversing transformations
  3. Recognize when a 3D transformation is singular

Suggested Activities

  1. Compare 2D and 3D: How does area scaling in 2D relate to volume scaling in 3D?
  2. Rotation exploration: Why do rotation matrices have det = 1?
  3. Singular detection: What happens to the parallelepiped when columns become coplanar?

Assessment Questions

  1. If det(A) = 8, how does a 1-cubic-unit region transform?
  2. What is the determinant of a 90° rotation matrix in 3D?
  3. Why does a scaling matrix [[2,0,0],[0,3,0],[0,0,4]] have det = 24?

References