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weelookang@gmail.com

Briefing Document: Vector Components Simulation Model

1. Overview

This document reviews the "Student Learning Space master quiz Vector Components QUIZ JavaScript HTML5 Applet Simulation Model," an interactive online tool designed to help students understand vector components. It's part of the Open Educational Resources / Open Source Physics @ Singapore project, aiming to provide engaging, interactive resources for physics education. The model uses JavaScript and HTML5, allowing for easy embedding and accessibility across devices.

2. Core Concepts

The primary focus is on teaching students about vectors and their decomposition into components:

• Vectors Defined: The resource emphasizes that a vector is a quantity possessing both magnitude and direction. It explicitly lists examples such as "displacement, velocity, acceleration, and force."
• Vector Components: The core idea is introduced: "a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes." The analogy of a vector pointing northwest having a "northward part and a westward part" helps in building intuition.
• Decomposition (Resolution): The model focuses on resolving a vector into its perpendicular components, usually along the x and y axes. This decomposition is described as "projections of the vector on a set of mutually perpendicular reference axes (basis vectors)."
• Mathematical Representation: The document explicitly provides the core equations used for the model:
• Component Calculation: ( A_{x} = A cos( \theta) ) and ( A_{y} = A sin( \theta) ) , where A is the magnitude of the vector and θ is the angle relative to the positive x-axis.
• Magnitude Calculation: ( |A| = \sqrt {( A_{x}^{2}+A_{y}^{2})} )
• Angle Calculation: ( \theta = tan^{-1} (\frac {A_{y}}{A_{x}}) ) , with conversions to degrees noted

3. Learning Objectives and Outcomes

The resource aligns with specific learning objectives and outcomes, referencing the Singaporean A-Level Physics curriculum:

• Learning Objectives: The document links to https://sites.google.com/moe.edu.sg/a-level-physics-tlg/foundations-of-physics/quantities-measurement which likely outlines broader measurement principles.
• Learning Outcomes: It specifically mentions "H2 - 1(h)(i)(j)", suggesting these are specific learning objectives in the H2 level of the A-Level curriculum, likely related to understanding and using vectors.
• Curriculum Emphasis: The resource is noted to focus on "Ways of Thinking and Doing [WOTD]" , indicating a focus on developing scientific thinking processes.
• Explicit Learning Goals: The document also lists sample learning goals such as: "(e) state what is meant by scalar and vector (how to resolve) quantities and give common examples of each" and "(f) calculate vector by means of components."

4. Simulation and Model Features

The core element is the interactive simulation that allows students to:

• Generate Random Vectors: Students can "click reset to generate a new random vector A of length |A| at an angle of theta to the positive x axis," providing varied practice.
• Calculate Components: The learning activity is for students to calculate the x and y components of the displayed vector using the formulas, reinforcing their practical application.
• Visual Exploration: There are interactive elements where when "Ax is selected, the PLAY button animates the effects of changing Ax alone" and the same for Ay, allowing for visual learning of how these components affect a vector.
• User-Defined Vectors: Students can experiment with custom values, allowing "this model can be used to calculate any length |A| by keying in the values."
• Self-Assessment: The model encourages practice with the statement, "test yourself whether you have enough practice calculating Ax and Ay."

5. Key Quotes

• "A vector is a quantity that has both magnitude and direction."
• "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."
• "As explained above a vector is often described by a set of vector components that add up to form the given vector."
• "The length or magnitude or norm of the vector a is denoted by |A|."

6. Intended Use and Pedagogical Approach

The simulation appears to be designed for:

• Active Learning: Students actively calculate vector components and test their understanding.
• Visual Learning: The simulation allows visualization of how components contribute to a vector.
• Repetitive Practice: The reset feature facilitates multiple calculations with new random vectors.
• Self-Directed Learning: Students are encouraged to "explore your own vector(s) here to test your understanding."
• Formative Assessment: The "test yourself" phrasing indicates the tool is designed for student self assessment and practice as opposed to a formal assessment.
• Integration with Learning Management Systems: The ability to embed the model via an iFrame suggests integration into platforms like the Student Learning Space (SLS).

7. Supplementary Resources The document also includes links to other simulations such as "Example JavaScript Model Ill 3.1: Vector Decomposition," and a Geogebra resource. There are links to similar simulations made by different authors, as well as links to other topics which include:

• "Mass and Spring with Accelerometer JavaScript HTML5 Applet Simulation Model"
• "AC or DC Appliances JavaScript Simulation Applet HTML5"
• "Bar Magnet Field Line Simulator JavaScript Simulation Applet HTML5"
• "Secondary Kinematics in Y direction or Free Fall JavaScript HTML5 Applet Simulation Model"
• "Horizontal 3D Circular Motion of Mass on a Table JavaScript WebGL Model"

8. Technical Details and Credits

• Platform: The simulation uses "JavaScript HTML5," making it widely accessible on web browsers.
• Credits: The resource acknowledges the developers and translators, which is important for open educational resources.
• Licensing: The content is licensed under "Creative Commons Attribution-Share Alike 4.0 Singapore License."
• Open Source: The document links to source code repositories, indicating this is part of an open source educational initiative.

9. Conclusion

The "Student Learning Space master quiz Vector Components QUIZ JavaScript HTML5 Applet Simulation Model" appears to be a well-designed, interactive, and pedagogically sound resource for teaching vector components. Its alignment with curriculum standards, focus on active learning, and provision of mathematical equations make it a valuable tool for physics education. The embedded nature, along with clear learning goals, positions it well for effective use within the Student Learning Space. The supplementary resources can also be used to teach related content.

Vector Components Study Guide

Quiz

Instructions: Answer the following questions in 2-3 sentences each.

1. What is a vector quantity and how does it differ from a scalar quantity?
2. Provide three examples of physical quantities that are vectors.
3. Explain why angular displacement and electric current are NOT vectors, even though they have magnitude and direction.
4. What are vector components?
5. When decomposing a vector into its components, what is the typical set of reference axes used?
6. Describe in words how you calculate the x-component (Ax) of a vector A given its magnitude |A| and the angle θ it makes with the x-axis?
7. Describe in words how you calculate the y-component (Ay) of a vector A given its magnitude |A| and the angle θ it makes with the x-axis?
8. Explain how to determine the magnitude |A| of a vector if you know its components Ax and Ay.
9. How do you calculate the angle a vector makes with the x-axis if you know the components Ax and Ay?
10. In the simulation model, what happens when the Ax or Ay checkboxes are selected and the play button is pressed?

Quiz Answer Key

1. A vector quantity has both magnitude and direction, and it also follows the rules of vector addition. In contrast, a scalar quantity has only magnitude and no direction.
2. Three examples of vector quantities include velocity, force, and displacement. These quantities require both a size and a direction to fully describe them.
3. Although angular displacement and electric current have magnitude and direction, they don't follow the rules of vector addition. This makes them non-vectors; they do not follow the commutative law of vector addition.
4. Vector components are the projections of a vector onto a set of mutually perpendicular reference axes. These components add together to form the original vector.
5. The typical set of reference axes used for vector decomposition are the x (horizontal) and y (vertical) axes. These axes are perpendicular to each other.
6. The x-component (Ax) of a vector is calculated by multiplying the vector's magnitude |A| by the cosine of the angle θ that the vector makes with the positive x-axis. The formula is Ax = |A|cos(θ).
7. The y-component (Ay) of a vector is calculated by multiplying the vector's magnitude |A| by the sine of the angle θ that the vector makes with the positive x-axis. The formula is Ay = |A|sin(θ).
8. The magnitude |A| of a vector is calculated using the Pythagorean theorem, which states that |A| is the square root of the sum of the squares of its components Ax and Ay or |A| = √ (Ax² + Ay²).
9. The angle a vector makes with the x-axis is calculated using the inverse tangent of the ratio of the y-component (Ay) to the x-component (Ax). This is denoted as θ = tan⁻¹(Ay/Ax).
10. When the Ax or Ay checkboxes are selected and the play button is pressed in the simulation, it animates the effect of changing the corresponding component (Ax or Ay) alone, showing visually how each component contributes to the overall vector.

Essay Questions

Instructions: Answer each question in essay format, demonstrating a comprehensive understanding of the topic.

1. Discuss the importance of vectors in physics, providing various examples of how they are used and why a scalar quantity is insufficient to describe these phenomena. Include real-world examples to strengthen your explanation.
2. Explain the concept of vector decomposition (resolving) and the significance of using components. Use the provided formulas to support your explanation. Furthermore, discuss the limitations of the model and suggest ways the model could be improved.
3. Compare and contrast the use of vector components to represent the same vector in different coordinate systems, providing examples of how they can be converted in different situations. Explain why it is important to maintain the magnitude and direction of a vector while decomposing it.
4. Analyze how the interactive simulation enhances understanding of vector components. Discuss specific interactions that aid in visualizing the process of vector decomposition, such as selecting checkboxes and using the play button.
5. Using your knowledge of both physics and trigonometry, explain the mathematics behind vector decomposition, the equations used in the simulation, and how to calculate vector components. Include a discussion of how the equations are derived and how they can be applied to real-world scenarios.

Glossary of Key Terms

• Vector: A quantity that has both magnitude and direction, and follows the rules of vector addition.
• Scalar: A quantity that has magnitude only and no direction.
• Magnitude: The size or length of a vector.
• Direction: The orientation of a vector in space.
• Vector Components: Projections of a vector onto a set of mutually perpendicular reference axes.
• Decomposition: The process of breaking a vector down into its components.
• Resolution: Another term for vector decomposition.
• Reference Axes: A set of coordinate axes (e.g., x and y axes) used for measuring the vector components.
• Resultant Vector: The vector sum of two or more vectors (often achieved by combining vector components).
• θ (theta): The symbol commonly used to represent the angle that a vector makes with the positive x-axis.
• |A|: The symbol used to denote the magnitude of a vector A.
• Ax: The x-component of vector A, representing the vector's projection on the x-axis.
• Ay: The y-component of vector A, representing the vector's projection on the y-axis.
• tan⁻¹: The inverse tangent function, used to calculate an angle from a ratio.

 

Learning Objectives: https://sites.google.com/moe.edu.sg/a-level-physics-tlg/foundations-of-physics/quantities-measurement

Learning Outcomes: H2 - 1(h)(i)(j)

Curriculum Emphasis: Ways of Thinking and Doing [WOTD https://vle.learning.moe.edu.sg/mrv/moe-library/lesson/view/c788167e-5865-4166-b147-89330b50b6b2/cover  SLS lesson includes questions that can be repeated indefinitely with random values inserted. 

 

Topics

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. For example, a vector that is directed northwest can be thought of as having two parts - a northward part and a westward part. A vector that is directed upward and rightward can be thought of as having two parts - an upward part and a rightward part.

Engage:

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is velocity, the magnitude of which is speed. For example, the velocity 4 meters per second upward could be represented by the vector $A_{y}$ = (0,4) (in 2 dimensions with the positive y axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement, linear acceleration, angular acceleration, linear momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field. Examples of quantities that have magnitude and direction but fail to follow the rules of vector addition: Angular displacement and electric current. Consequently, these are not vectors.

Component:

As explained above a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set. For simplicity, let's assume x and y direction as the reference axes, referring to the model above.

the equations for the components are:

 $A_{x} = A cos( \theta)$

 $A_{y} = A sin( \theta)$

 

Model:

This model uses the following equations:

Length or Magnitude |A|:

The length or magnitude or norm of the vector a is denoted by |A|.

 $|A| = \sqrt {( A_{x}^{2}+A_{y}^{2})}$

Angle:

The angle in radian of the vector that makes with the positive x axis is

$\theta = tan^{-1} (\frac {A_{y}}{A_{x}})$

the conversion to degree is

$\theta_{degree} = (tan^{-1} (\frac{A_{y}}{A_{x}}) )( \frac{180}{\pi})$

or

$\theta_{degree} = ( \theta )( \frac{180}{\pi})$

in the design of the slider values which shows in degree, the equation that converts back from degree to radian is used

$\theta = ( \theta_{degree} )( \frac{\pi}{180})$

Calculate the components of the vector A

1. click reset to generate a new random vector A of length |A| at an angle of theta to the positive x axis.
2. use the formula Ax = |A|cos(theta) to find the x direction component Ax.
3. use the formula Ay = |A|sin(theta) to find the y direction component Ay.
4. click reset to random generate a new vector A.
5. 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 = 63 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 herehttp://weelookang.blogspot.sg/2014/10/components-vector-model.html 

Sample Learning Goals

(e) state what is meant by scalar and vector(how to resolve) quantities and give common examples of each

 

Version:

1. http://weelookang.blogspot.sg/2014/10/components-vector-model.html

2. http://weelookang.blogspot.sg/2014/10/components-vector-model.html CLICK https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_horizontalvertical/horizontalvertical_Simulation.xhtml author: lookang, Fu-Kwun, Andreu Glasmann, Wolfgang Christian, and Mario Belloni offline use: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_horizontalvertical.zip source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_horizontalvertical.zip author of EJS 5: Paco.

3. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=662.0 Represent a vector as horizontal and vertical components applet by Loo Kang Wee and Fu-Kwun Hwang
4. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=661.0 Magnitude of a Vector by Loo Kang Wee and Fu-Kwun Hwang

 

Other Resources

1. Example JavaScript Model Ill 3.1: Vector Decomposition written by Andreu Glasmann, Wolfgang Christian, and Mario Belloni 
2. https://www.geogebratube.org/student/m38272 

Frequently Asked Questions on Vectors and Vector Components

1. What is a vector and how does it differ from a scalar? A vector is a quantity that has both magnitude (size) and direction. Common examples include displacement, velocity, acceleration, and force. In contrast, a scalar quantity has only magnitude, such as time or mass. Vectors follow specific rules of addition. Quantities that have magnitude and direction but do not adhere to these rules are not vectors.
2. Why are vectors important in physics? Vectors are fundamental because they allow us to represent and analyze physical quantities that have both a size and a direction. This is crucial in understanding motion, forces, fields, and many other phenomena in physics. Vector representation allows us to model real-world phenomena and perform calculations that accurately describe their behavior.
3. What are vector components and why do we use them? Vector components are the projections of a vector onto a set of mutually perpendicular reference axes (usually x and y). Breaking a vector into its components allows us to analyze it by considering each direction independently. This simplifies many calculations and problems involving vectors at angles. For example a force applied at an angle could be thought of as having a horizontal and a vertical component which each contribute independently to the overall effect.
4. How do you calculate the components of a vector? If a vector 'A' has a magnitude of |A| and makes an angle θ with the positive x-axis, then its x-component (Ax) is calculated as Ax = |A| * cos(θ), and its y-component (Ay) is calculated as Ay = |A| * sin(θ). These formulas use trigonometry to resolve the vector into its horizontal and vertical pieces. The angle can be in radians or degrees, but must be consistent.
5. How do you calculate the magnitude and angle of a vector from its components? Given the x-component (Ax) and y-component (Ay), the magnitude of the vector 'A' is found using the Pythagorean theorem: |A| = sqrt(Ax² + Ay²). The angle θ that the vector makes with the positive x-axis can be found using θ = tan⁻¹(Ay/Ax). The resulting angle might need adjusting depending on what quadrant the vector is in.
6. Can a vector have components of zero? Yes, a vector can have zero components. For example, a vector lying directly along the x-axis will have a y-component equal to zero, and vice versa. Also, a zero vector (a vector with zero magnitude) will have both x and y components equal to zero.
7. What is the difference between a vector being 'resolved' and 'decomposed'? The terms 'resolved' and 'decomposed' are often used interchangeably when discussing vectors. Both terms refer to the process of breaking down a vector into its components along a set of coordinate axes. There is not a meaningful distinction between the two.
8. What kind of real-world applications use the principles of vector decomposition? The concept of vector decomposition has very many real-world applications. It is used in engineering, construction, and physics, including understanding forces in structures, projectile motion, navigation, and even analyzing electrical and magnetic fields. For example, calculating the forces acting on a bridge, or analyzing the motion of an object through the air, involves resolving vectors into their components.