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Decomposition of (invariant or moving along the axis ) Vector along any 2 perpendicular axes Advance JavaScript HTML5 Applet Simulation Model from NJC

 

 

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Measurement of length and time

Description

A vector is a quantity that has both magnitude and direction. Displacement, velocity, acceleration, and force are the vector quantities. in simpler examples, vectors were simply directed up, down, left or right. In situations in which vectors are directed at angles to the customary x(horizontal)-y(vertical) coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes. What if the customary x-y coordinate axes is not (horizontal) and (vertical)? Sometimes, it could be useful to resolve vectors along and perpendicular to a slope, direction of travel than always horizontal and vertical,

Calculate the components of the vector A in x' and y' direction

1. rotate the x' axis by dragging the GREEN slider to the 2 perpendicular direction to be resolved.
2. change the vector A to be resolved by dragging the BLACK |A| slider and the CYAN angle slider.
3. click reset to generate a new random vector A of length |A| at an angle of theta to the positive x' axis.
4. use the formula Ax' = |A|cos(theta) to find the x' direction component Ax.
5. use the formula Ay' = |A|sin(theta) to find the y' direction component Ay.
6. click reset to random generate a new vector A.
7. test yourself whether you have enough practice calculating Ax' and Ay'.

Other interesting fun activities

1. this model can be used to calculate any length |A| by keying in the values such as |A| =500 and theta = 53 degrees. Select the checkboxes Ax' and Ay' to see the new answers.
2. when Ax' is selected, the PLAY button animates the effects of changing Ax' alone.
3. when Ay' is selected, the PLAY button animates the effects of changing Ay' alone.
4. explore your own vector(s) here to test your understanding.
5. leave me a Google+ comment/question/requests etc here http://weelookang.blogspot.com/2014/10/decomposition-of-vector-advance-model.html

Sample Learning Goals

(e) state what is meant by scalar and vector (how to resolve) quantities and give common examples of each
(f) calculate vector by means of components in non-horizontal and vertical axes.

Version:

  1. http://weelookang.blogspot.com/2014/10/decomposition-of-vector-advance-model.html


 

Translations

Code Language Translator Run

Credits

weelookang@gmail.com; Fu-Kwun Hwang; Tat Leong Lee

 

Key Concepts and Ideas:

  • Definition of a Vector: The document starts by defining a vector as "a quantity that has both magnitude and direction," providing examples like displacement, velocity, acceleration, and force. This establishes the fundamental concept that the simulation builds upon.
  • Quote: "A vector is a quantity that has both magnitude and direction."
  • The Need for Vector Decomposition: It highlights that while simple examples might involve vectors directed along standard axes, many real-world scenarios involve vectors at angles. This necessitates the "mathematical trick" of decomposition.
  • Quote: "In situations in which vectors are directed at angles to the customary x(horizontal)-y(vertical) coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes."
  • Generalization to Non-Standard Axes: A crucial point is that the simulation allows decomposition along any two perpendicular axes, not just horizontal and vertical. This is particularly useful for analyzing motion along inclined planes or directions of travel.
  • Quote: "Sometimes, it could be useful to resolve vectors along and perpendicular to a slope, direction of travel than always horizontal and vertical,"
  • Interactive Simulation Features: The document details how users can interact with the simulation to explore vector decomposition:
  • Rotating Axes: Users can "rotate the x' axis by dragging the GREEN slider to the 2 perpendicular direction to be resolved." This allows visualization of decomposition along arbitrary perpendicular directions.
  • Modifying the Vector: Users can "change the vector A to be resolved by dragging the BLACK |A| slider and the CYAN angle slider." This enables exploration of how magnitude and direction affect the components.
  • Random Vector Generation: Clicking "reset" generates "a new random vector A of length |A| at an angle of theta to the positive x' axis," providing varied scenarios for practice.
  • Component Calculation: The simulation encourages users to "use the formula Ax' = |A|cos(theta) to find the x' direction component Ax" and "use the formula Ay' = |A|sin(theta) to find the y' direction component Ay." This reinforces the mathematical relationship between the vector and its components.
  • Visualizing Components: Selecting the checkboxes for Ax' and Ay' reveals the calculated component values.
  • Animating Component Effects: The "PLAY" button, when Ax' or Ay' is selected, "animates the effects of changing Ax' alone" or "animates the effects of changing Ay' alone," respectively. This helps visualize the independent contribution of each component.
  • Custom Vector Input: Users can "calculate any length |A| by keying in the values such as |A| =500 and theta = 53 degrees" to analyze specific vector scenarios.
  • Learning Goals: The simulation aims to help students:
  • "(e) state what is meant by scalar and vector (how to resolve) quantities and give common examples of each"
  • "(f) calculate vector by means of components in non-horizontal and vertical axes."
  • Open Educational Resource: The model is presented as an "Open Educational Resource / Open Source Physics @ Singapore," emphasizing its accessibility and potential for integration into educational settings.
  • Technical Details: The simulation is an "Advance JavaScript HTML5 Applet Simulation Model" that can be embedded in webpages via an iframe. It is designed to run on various platforms, including desktops, laptops, tablets, and smartphones. The second source explicitly states it was "Compiled with EJS 6.0 (191124)."
  • Credits and Versions: The developers are credited as weelookang@gmail.com; Fu-Kwun Hwang; and Tat Leong Lee. The document also lists various versions and related resources, indicating the evolution and broader context of this simulation within a collection of physics education tools.
  • Integration with Learning Platforms: The mention of "SLS MOE Calendar Planner" and "SLS components" in the "Other Resources" section suggests efforts to integrate these interactive simulations into national learning management systems.
  • Broader Scope of Resources: The extensive list of "Other Resources" indicates a rich collection of interactive physics and mathematics simulations available from the same source, covering a wide range of topics. This positions the vector decomposition model within a larger context of digital educational tools.

Most Important Ideas/Facts:

  • The simulation provides an interactive and visual way to understand vector decomposition beyond standard horizontal and vertical axes.
  • It allows users to manipulate the vector and the orientation of the perpendicular axes to observe the resulting components.
  • The simulation reinforces the mathematical formulas used for vector decomposition (using sine and cosine).
  • It caters to different learning styles through interactive manipulation, visual feedback, and opportunities for practice.
  • It is an open educational resource, making it freely available for educational use.
  • The model is part of a larger suite of interactive physics simulations developed by Open Source Physics @ Singapore.

In Conclusion:

The "Decomposition of Vector (invariant or moving along the axis) ... Advance JavaScript HTML5 Applet Simulation Model" is a valuable educational tool for teaching vector decomposition. Its interactive features, focus on general perpendicular axes, and clear learning goals make it a potentially effective resource for students learning introductory physics concepts. The context provided by the extensive list of other resources highlights a broader commitment to developing and sharing open-source interactive simulations for science and mathematics education.

 

 

Study Guide: Vector Decomposition

I. Key Concepts:

  • Scalar Quantities: Define what a scalar quantity is and provide at least three examples from everyday life or physics (other than those explicitly mentioned as vector examples).
  • Vector Quantities: Define what a vector quantity is, emphasizing the difference from scalar quantities, and list at least three examples from the text (Displacement, Velocity, Acceleration, Force).
  • Magnitude and Direction: Explain why both magnitude and direction are crucial for defining a vector.
  • Coordinate Axes: Describe the purpose of coordinate axes (customary x-y, and generalized x'-y') in representing vectors.
  • Vector Decomposition: Explain the process of vector decomposition. Why is it a useful mathematical trick when dealing with vectors at angles?
  • Perpendicular Components: Define what is meant by resolving a vector into two perpendicular components. Explain why these components are often chosen along the coordinate axes.
  • Trigonometric Relationships: Explain the role of sine and cosine functions in determining the components of a vector. Specifically, how are the magnitudes of the x' and y' components (Ax' and Ay') related to the magnitude of the original vector (|A|) and the angle (theta)?
  • Generalization of Axes: Explain the benefit of resolving vectors along axes that are not strictly horizontal and vertical (e.g., along and perpendicular to a slope or direction of travel).
  • Simulation Tool Functionality: Describe how the provided simulation tool can be used to:
  • Rotate the coordinate axes.
  • Adjust the magnitude and direction of a vector.
  • Visualize the components of a vector.
  • Calculate the components using the given formulas.
  • Explore the effects of changing individual components.

II. Quiz:

  1. What is the fundamental difference between a scalar and a vector quantity? Provide one unique example of each that was not explicitly listed in the provided text.
  2. Explain why decomposing a vector into its perpendicular components is a useful technique when analyzing physical situations. Give a brief example of a scenario where this would be helpful.
  3. In the context of vector decomposition using trigonometry, what does the magnitude |A| represent, and what does the angle theta represent? How are these related to the calculated components Ax' and Ay'?
  4. Why might it be more useful in some scenarios to resolve a vector along axes that are not horizontal and vertical? Provide a hypothetical example where this would simplify the problem.
  5. Describe two interactive features of the simulation tool mentioned in the text and explain how they can aid in understanding vector decomposition.
  6. Using the formulas provided, if a vector A has a magnitude of 10 units and is at an angle of 30 degrees to the x' axis, what are the approximate magnitudes of its x' and y' components? (You do not need a calculator, just recall the approximate values of sin(30) and cos(30)).
  7. What happens to the y' component of a vector if the angle theta between the vector and the x' axis is 0 degrees? Explain your reasoning using the relevant trigonometric function.
  8. The text mentions "invariant or moving along the axis" in the title. In the context of the simulation and vector decomposition, what does "invariant" likely refer to regarding the vector?
  9. State one of the sample learning goals mentioned in the text related to vectors and briefly explain how understanding vector decomposition contributes to achieving that goal.
  10. What are the formulas used to calculate the x' direction component (Ax') and the y' direction component (Ay') of a vector A with magnitude |A| and angle theta relative to the x' axis?

III. Quiz Answer Key:

  1. A scalar quantity has only magnitude, while a vector quantity has both magnitude and direction. Example scalar: temperature; Example vector: the wind velocity (e.g., 15 mph to the north).
  2. Decomposing a vector simplifies analysis by allowing us to consider the effects of the vector along independent perpendicular directions. For example, when analyzing projectile motion, we can analyze the horizontal and vertical motions separately.
  3. |A| represents the length or size of the vector, and theta represents the angle the vector makes with the positive x' axis. Ax' = |A|cos(theta) and Ay' = |A|sin(theta) relate these quantities.
  4. Resolving along non-horizontal/vertical axes can align the components with the physical constraints or motion of a system, simplifying calculations. For instance, analyzing motion on an inclined plane is easier when resolving gravity into components parallel and perpendicular to the slope.
  5. One feature is the ability to drag sliders to change the vector's magnitude and angle, allowing for direct observation of how the components change. Another is the "PLAY" button that animates the effect of changing a single component, isolating its contribution.
  6. Approximately, cos(30) is 0.87 and sin(30) is 0.5. Therefore, Ax' ≈ 10 * 0.87 = 8.7 units, and Ay' ≈ 10 * 0.5 = 5 units.
  7. The y' component will be zero. This is because Ay' = |A|sin(0), and sin(0) equals 0. Thus, there is no component of the vector in the y' direction when it lies along the x' axis.
  8. "Invariant" likely refers to a vector whose magnitude and direction remain constant during the decomposition process or within a specific scenario being modeled.
  9. One learning goal is to "calculate vector by means of components in non-horizontal and vertical axes." Understanding decomposition is fundamental to performing this calculation, as it provides the method for finding these components.
  10. The formula for the x' direction component is Ax' = |A|cos(theta), and the formula for the y' direction component is Ay' = |A|sin(theta).

IV. Essay Format Questions:

  1. Discuss the importance of distinguishing between scalar and vector quantities in physics. Provide examples of how failing to account for the directional nature of vectors can lead to incorrect analysis or predictions in real-world scenarios.
  2. Explain the mathematical process of resolving a vector into its perpendicular components. Detail the role of trigonometry in this process and discuss why this technique is a fundamental tool in the study of physics and engineering.
  3. Consider a situation where a force vector is acting on an object moving along an inclined plane. Analyze why resolving this force into components parallel and perpendicular to the plane would be a more useful approach than using standard horizontal and vertical components.
  4. Critically evaluate the benefits of using interactive simulation tools, like the one described in the text, for learning about vector decomposition. How do these tools enhance understanding compared to traditional methods of instruction?
  5. The text mentions that vector decomposition is a useful "mathematical trick." Discuss whether you agree with this characterization. Explain why this process is either merely a trick or a fundamental concept that reflects the nature of vector quantities.

V. Glossary of Key Terms:

  • Vector: A quantity that has both magnitude (size) and direction. Examples include displacement, velocity, and force.
  • Scalar: A quantity that has only magnitude and no direction. Examples include temperature, mass, and time.
  • Magnitude: The size or length of a vector, always a non-negative scalar quantity.
  • Direction: The orientation of a vector in space, often specified by an angle relative to a reference axis.
  • Component of a Vector: The projection of a vector onto a specific axis. When a vector is resolved into perpendicular components, these components are vectors themselves, directed along the chosen axes, whose vector sum equals the original vector.
  • Coordinate Axes: A set of perpendicular lines (e.g., x and y axes) used as a frame of reference for describing the position and direction of vectors.
  • Decomposition (or Resolution) of a Vector: The process of breaking down a single vector into two or more component vectors that, when added together, produce the original vector. Often, these components are chosen to be perpendicular to each other.
  • Trigonometry: The branch of mathematics dealing with the relationships between the sides and angles of triangles, particularly right-angled triangles. The sine (sin), cosine (cos), and tangent (tan) functions are crucial for vector decomposition.
  • Angle (Theta - θ): In the context of vector decomposition, this typically refers to the angle between the vector and a chosen reference axis, such as the positive x' axis.
  • Simulation: A computer-based model of a real-world system or process, allowing users to interact with and explore its behavior. In this case, a simulation designed to visualize and calculate the components of a vector.

 

Versions:

  1. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=634.0 sine and cosine by Fu-Kwun Hwang
  2. https://sg.iwant2study.org/ospsg/index.php/interactive-resources/physics/01-measurements/822-slsvectorsum04 
  3. https://weelookang.blogspot.com/2019/01/decomposition-of-invariant-or-moving.html 

 

Other Resources

 

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Frequently Asked Questions: Vector Decomposition

1. What is a vector quantity, and how does it differ from a scalar quantity?

A vector quantity is one that possesses both magnitude (size) and direction. Examples provided include displacement, velocity, acceleration, and force. In contrast, a scalar quantity has only magnitude and no direction (examples of scalars are not explicitly given in this text, but could include mass, temperature, and time).

2. What does it mean to decompose a vector?

To decompose a vector means to break it down into two or more component vectors that, when added together, produce the original vector. The source material specifically focuses on decomposing a vector into two perpendicular components.

3. Why is vector decomposition a useful technique in physics?

Vector decomposition is useful in situations where a vector is directed at an angle to a chosen coordinate system (not necessarily just horizontal and vertical). By resolving a vector into components along these axes, it simplifies mathematical analysis and calculations, allowing us to treat each component independently. This is particularly helpful when dealing with motion along a slope or any direction other than the standard horizontal and vertical.

4. How can I calculate the components of a vector A along two perpendicular axes (x' and y') using the provided simulation model?

The simulation allows you to visually and mathematically determine the components. First, you can rotate the x' axis to the desired perpendicular direction using a green slider. Then, you can adjust the magnitude (|A|) and angle (theta) of the vector A using black and cyan sliders, respectively. The simulation then provides the formulas to calculate the components: Ax' = |A|cos(theta) for the x' direction and Ay' = |A|sin(theta) for the y' direction, where theta is the angle with respect to the x' axis.

5. Can the simulation model be used for purposes beyond calculating vector components?

Yes, the model has additional functionalities. You can input specific values for the magnitude (|A|) and angle (theta) to calculate the corresponding components. Furthermore, by selecting the checkboxes for Ax' or Ay' and pressing the PLAY button, you can observe the effect of changing each component independently through animation.

6. What if the axes along which I want to decompose the vector are not horizontal and vertical?

The simulation explicitly addresses this scenario. It allows you to rotate the coordinate axes (x' and y') to any two perpendicular directions using the green slider. This enables you to resolve vectors along axes that might be aligned with a slope or a direction of travel, rather than being strictly horizontal and vertical.

7. Are there any learning goals associated with this simulation model?

Yes, the sample learning goals include understanding the difference between scalar and vector quantities and being able to calculate vector components along non-horizontal and vertical axes. The simulation is designed to provide practice in these areas.

8. Where can I find more information or provide feedback regarding this simulation?

The "About" section provides contact information (weelookang@gmail.com) and a link to a blog (http://weelookang.blogspot.com/2014/10/decomposition-of-vector-advance-model.html) where you can leave comments, questions, or requests. The "Credits" section also lists contributors.

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Category: 01 Measurements
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