Teacher Demonstration
Use the live model as a shared screen demonstration before students try their own predictions and observations.
Measurement
Decomposition Of Vector (Any Mutually Perpendicular) Advance Model
Explore Decomposition Of Vector (Any Mutually Perpendicular) Advance Model as an interactive EJS simulation for Measurement.
1. Watch or Launch
Launch the Interactive
Open the simulation, adjust the controls, and compare what changes on screen before answering the concept-check questions.
2. Big Ideas
Key idea This model resolves one vector along a rotated pair of mutually perpendicular axes. Students choose the x' direction, set the vector magnitude and angle, then use cosine and sine to calculate Ax' and Ay'. The learning point is that components depend on the chosen axes, not always on the screen's horizontal and vertical directions.
What Students Can Learn
- A vector can be decomposed along any two mutually perpendicular axes.
- The x' axis can be rotated to suit a slope or direction of travel.
- When theta is measured from x', the component along x' is Ax' = |A|cos(theta).
- The perpendicular component is Ay' = |A|sin(theta), with direction and sign checked against the diagram.
Guiding Question
How do the chosen x' and y' axes change the component values for the same vector A?
3. Try the Investigation
Set the Rotated Axes
Use the green slider to choose the x' axis direction. The y' direction remains perpendicular to x'.
Choose Vector A
Use the black |A| slider and cyan angle slider to set the vector magnitude and theta measured from the positive x' axis.
Calculate Components
Compute Ax' = |A|cos(theta) and Ay' = |A|sin(theta), then compare with the displayed projections.
Reset and Test
Use reset to generate a new vector and check whether your formula, angle reference, and chosen axes stay consistent.
4. Teacher Notes
Lesson Use
Use this page when students need to resolve forces, displacements, or other vectors along a slope or direction of travel instead of default horizontal and vertical axes.
Discussion Prompts
Ask: Which axis is theta measured from? Why can we rotate the axes? When would sine and cosine swap because the reference angle changed?
Teaching Moves
Require students to draw and label x' and y' before calculating. Then ask them to explain the sign and direction of each component, not only its magnitude.
Model Notes
The source intro names the green slider for rotating x', the black |A| slider for magnitude, the cyan angle slider for theta, reset for random practice, and Ax'/Ay' formulas using cosine and sine.
5. Concept Check
These questions are generated from the topic and the concept illustrated by the simulation. Use them after students have explored the model.
Concept Score
Correct first attempts build a streak and unlock higher point multipliers on this device.
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Answer each question once to build your streak.
1. What does the green slider change in this model?
2. If theta is measured from the positive x' axis, what is Ax'?
3. If theta is measured from the positive x' axis, what is Ay'?
4. Why are the components not always simply horizontal and vertical?
5. What should a strong answer include?
Expert Challenge
Unlocks after 3 correct concept-check answers on this page.
1. The x' axis is rotated to follow a slope, and vector A makes angle theta from positive x'. Which calculation correctly finds the component along the slope?
2. A student calculates components correctly but uses the usual horizontal and vertical axes after the model's x' axis has been rotated. What is the best feedback?
3. If theta were accidentally measured from y' instead of x', which expert warning is most useful?
4. Why is reset useful in this simulation after one worked example?
5. Which explanation best shows understanding of decomposition in this model?
7. Learning Pulse
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