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3D Volume Scaling Interactive Visualizer

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/volume-scaling-3d/main.html" height="392px" scrolling="no"></iframe>

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

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