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Vector Operations Playground

Run the Vector Operations Playground Fullscreen

About This MicroSim

This interactive playground allows students to explore vector operations by directly manipulating vectors and observing the results in real-time. Students can perform vector addition, subtraction, and scalar multiplication while seeing both the geometric and numerical representations.

Learning Objective: Students will apply vector addition, subtraction, and scalar multiplication by manipulating vectors interactively and predicting results before seeing them visualized.

How to Use

  1. Drag Vectors: Click and drag the circular endpoints of vectors u (blue) and v (red) to position them anywhere on the grid
  2. Select Operation: Use the radio buttons to choose between:
  3. Add: Shows u + v (green result vector)
  4. Subtract: Shows u - v (green result vector)
  5. Scalar ×: Shows c·u where c is controlled by the slider
  6. Adjust Scalar: When in scalar multiplication mode, use the slider to change the scalar value from -3 to 3
  7. Toggle Visualizations:
  8. Parallelogram: Shows the parallelogram construction for addition
  9. Components: Shows projection lines to the axes
  10. Animate: Click to see a step-by-step animation of the operation
  11. Reset: Return all vectors to their default positions

Key Concepts Demonstrated

  • Vector Addition: The parallelogram rule and tip-to-tail method
  • Vector Subtraction: Finding the difference vector u - v
  • Scalar Multiplication: How scalars stretch, shrink, or reverse vectors
  • Component Operations: How operations work on individual components

Mathematical Formulas

Addition: \(\mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y)\)

Subtraction: \(\mathbf{u} - \mathbf{v} = (u_x - v_x, u_y - v_y)\)

Scalar Multiplication: \(c\mathbf{u} = (c \cdot u_x, c \cdot u_y)\)

Embedding This MicroSim

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/vector-operations-playground/main.html"
        height="552px"
        width="100%"
        scrolling="no">
</iframe>

Lesson Plan

Grade Level

Undergraduate introductory linear algebra or advanced high school mathematics

Duration

20-25 minutes

Prerequisites

  • Understanding of 2D coordinate systems
  • Basic knowledge of vectors as arrows with magnitude and direction

Learning Activities

  1. Exploration (5 min): Let students freely drag vectors and observe how operations change
  2. Addition Investigation (5 min):
  3. Enable parallelogram view
  4. Ask students to verify the parallelogram rule geometrically
  5. Have them predict the sum before moving vectors
  6. Subtraction Investigation (5 min):
  7. Switch to subtraction mode
  8. Explore how u - v relates to the vector from v's tip to u's tip
  9. Scalar Multiplication (5 min):
  10. Vary the scalar from -3 to 3
  11. Observe what happens at c = 0, c = 1, c = -1
  12. Synthesis (5 min): Combine concepts to solve problems

Discussion Questions

  1. What is the geometric meaning of the parallelogram in vector addition?
  2. If u + v = w, what is u - v geometrically?
  3. What happens to the direction of a vector when multiplied by a negative scalar?
  4. Can two different pairs of vectors have the same sum?

Assessment Ideas

  • Given a target point, find u and v that add to reach it
  • Predict the result of an operation before seeing it
  • Find a scalar that makes cu equal to a specific vector

References

  1. 3Blue1Brown - Linear combinations, span, and basis vectors
  2. Khan Academy - Vector Addition
  3. Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. Chapter 1.