Vectors: Properties and Operations - Complete Toolkit

Objectives

1. To understand that a vector is a quantity that has a magnitude and direction and be able to graphically represent a vector using a scaled diagram and the counter-clockwise from east convention of direction.
2. To add vectors using a head-to-tail addition method and a scaled vector addition diagram and be able to identify the magnitude and direction of the resultant.
3. To add right-angle vectors, using the Pythagorean theorem to determine the resultant magnitude and trigonometric functions to determine the resultant direction.
4. To resolve vectors into components and to use the component method in order to add two or more non-perpendicular vectors in order to determine the resultant.
5. To understand that perpendicular components of motion are independent of each other and to use such an understanding to solve relative velocity problems such as riverboat problems.

Readings from The Physics Classroom Tutorial

1. The Physics Classroom Tutorial, Motion and Forces in Two Dimensions Chapter, Lesson 1
   
   http://www.physicsclassroom.com/class/vectors/Lesson-1/Vectors-and-Direction

Interactive Simulations

1. Boston University EJS: Vector Addition w/Integer Components
   
   Good choice to give beginners practice in 2D vector addition. It’s relatively simple, as the x and y components of each vector are integers. Given the magnitude and direction of two vectors, students determine the x and y components, the length of each, and resultant vector sum. Click “Check Answers” to get immediate feedback; click “Show Answers” to display correct responses. 
   
   
   
   
   
    
2. Boston University (EJS) Simulation:  Vector Addition and Pattern
   
   Kids will have fun with this simulation, which illustrates the tip-to-tail method of adding vectors. As students adjust lengths and angles of two vectors, the resultant vector and components are displayed.  BUT WAIT.  Students can also set rotation rates and watch the vectors spin, tracing out interesting patterns on the screen.
   
   
   
   
   
    
3. Boston University EJS - Vector Addition
   
   Students who have mastered vector addition with integers can move up to this simulation, which introduces vectors with non-integer lengths. 
   
   
   
   
   
    
4. PhET Simulation:  Vector Addition
   
   This interactive model lets students drag vectors onto a grid, change their length and angle, and sum them together. The magnitude, angle, and x/y components of each vector are automatically displayed in several formats. For beginning learners, we suggest using one or both of the PhET Teacher-Contributed Lessons (see links directly below)
    
   PhET Vectors Lab Guide
   http://phet.colorado.edu/en/contributions/view/3140
   This is a student guide created specifically for use with the simulation “Vector Addition”. It provides explicit instructions on how to use the simulation, then presents a short assignment for using the model to calculate resultant vectors. Allow about 30 minutes.  (86 kb Word document)
    
   Virtual Vector Lab
   http://phet.colorado.edu/en/contributions/view/3047
   This student guide, created by a high school teacher, provides structured guidance for students as they use the PhET simulation “Vector Addition”. It gives detailed directions for using the simulation and scaffolded support for the tip-to-tail method and the Pythagorean Theorem.  Allow two class periods.
   
   
   
   
   
5. The Riverboat Simulator
   
   This simulation from The Physics Classroom’s Shockwave Physics Studios allow the user to vary the speed and direction of a boat crossing a river and the river speed and width in order to explore the effect of a variable on the time to cross the river and the distance traveled downstream. Comes with a proposed activity sheet:  http://www.physicsclassroom.com/shwave/rboatdirns.cfm.
   
    

Video and Animations

Labs and Investigations

Demonstration Ideas

Minds On Physics Internet Modules:

The Minds On Physics Internet Modules are a collection of interactive questioning modules that target a student’s conceptual understanding. Each question is accompanied by detailed help that addresses the various components of the question.

1. Vectors and Projectile Module, Ass’t VP1 -  Direction of Vectors
2. Vectors and Projectile Module, Ass’t VP2 -  Vector Addition Diagrams
3. Vectors and Projectile Module, Ass’t VP3 -  Vector Addition Applications
4. Vectors and Projectile Module, Ass’t VP4 -  Adding Right Angle Vectors
5. Vectors and Projectile Module, Ass’t VP5 -  Vector Components
6. Vectors and Projectile Module, Ass’t VP6 -  Velocity and River Boats
   
   Link: http://www.physicsclassroom.com/mop

 
 

Concept Building Exercises:

1. The Curriculum Corner, Vectors and Projectiles, Vector Representation
2. The Curriculum Corner, Vectors and Projectiles, Addition of Vectors
3. The Curriculum Corner, Vectors and Projectiles, Vector Resolution and Vector Addition
4. The Curriculum Corner, Vectors and Projectiles, Vector Addition by Components
5. The Curriculum Corner, Vectors and Projectiles, Relative Velocity - Riverboat Problems
   
   Link: http://www.physicsclassroom.com/curriculum/vectors

Problem-Solving Exercises:

1. The Calculator Pad, Vectors and Projectiles, Problems #1 - #20
   
   Link: http://www.physicsclassroom.com/calcpad/vecproj

 

 

Real Life Connections:

1. CIESE Real-Time Data Project: Navigational Vectors
   
   Do your students ask, “When will we ever use this?”  This project-based unit can help bridge the gap between vector concepts and use of vector operations by real people in authentic situations. Activities (which can be parsed out) include reading real-time weather maps, tracking airplanes flying in U.S. skies, calculating vector components, analyzing effects of wind velocity, and completing training similar to a private pilot certification program.
   
   
   
    
2. NASA Dryden Flight Research Center:  Space Vectors
   
   This complete lesson was developed for pre-calc students who have completed a course in trigonometry. It introduces vectors in 3 dimensions, thus requires prior knowledge of spherical coordinates, azimuth, elevation, range, and vector notation. Students will use Google Earth to find their school, using NASA’s naming conventions for longitude & latitude. Then students will use spherical coordinates and vector addition to find position coordinates of the international space shuttle with respect to the Earth’s center and Dryden Flight Research Center.
    

Common Misconception:

1. The Order of Vector Addition Does Not Matter
   
   Students are often under the impression that the resultant of two or more vectors depends upon the order in which they are added. This misconception can be easily dispelled by simply having students measure out two right-angled vectors in the two different orders. The destination is the same despite the order in which they are added.
   
   
    
2. Perpendicular Components
   
   The independence of perpendicular components of motion is an important principle derived from a unit on vectors that will foster a better understanding of projectile motion. An alteration in a north-south component of motion will not affect an east-west component of motion. This principle is most easily reinforced by a study of the motion of a riverboat heading straight across a river. A change in the riverboat speed does not affect the distance that the boat travels down the river. In a similar manner, a change in the horizontal velocity of a projectile does not affect the time required for the projectile to fall to the ground. Perpendicular components of motion are independent of each other.

 

 

Related PER (Physics Education Research)

1. University of Monterrey Physics Education Research Group
   
   This article from the University of Monterrey Physics Education Research group discusses results of a large study gauging the understanding of vector concepts among 2,000+ college students.  Areas of strength included representing x and y components, calculating magnitude, finding vector sums, and calculation of dot and cross products using given equations. However, even after 3 courses in physics, the students showed surprising weakness in calculating vector direction, subtraction of vectors in 2D, and geometric interpretations of dot and cross products.
   
   
    
2. Physics Education Research Conference Proceedings Annotated Collection
   Student difficulties with vector concepts are often pervasive, extending persistently beyond high school through multiple undergraduate courses in physics and engineering. In the past two years, PER (physics education research) has focused more on error patterns and promising interventions. Click the link below for an annotated collection of the most relevant research publications we found for you. (All are in condensed formats and available for free download.)
   
   
   
   
    
3. PER Examples:  Vector Subtraction Interactive Homework Problem
   Great interactive problem to help students overcome difficulties with vector subtraction. It takes learners step-by-step through a conceptual analysis, then proceeds to a strategic analysis with explicit support for setting up equations appropriately. Finally, the learner will perform the math with scaffolded help available at each waypoint.  Questions are carefully crafted to promote success without spoon-feeding answers.
   
   
    

 

 

Standards:

A. Next Generation Science Standards (NGSS)

• High School: Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another.

• Use a model to predict the relationships between systems or between components of a system.
• Use a model to provide mechanistic accounts of phenomena.

• Plan and conduct an investigation individually and collaboratively to produce data to serve as the basis for evidence, and in the design: decide on types, how much, and accuracy of data needed to produce reliable measurements and consider limitations on the precision of the data (e.g., number of trials, cost, risk, time), and refine the design accordingly.

• Analyze data using tools, technologies, and/or models (e.g., computational, mathematical) in order to make valid and reliable scientific claims or determine an optimal design solution.
• Analyze data using computational models in order to make valid and reliable scientific claims.

• Use mathematical representations of phenomena to describe explanations.
• Create a computational model or simulation of a phenomenon, process, or system
• Use mathematical representations of phenomena to support and revise explanations

• Communicate scientific information about phenomena in multiple formats (including orally, graphically, textually, and mathematically)

 
B. Common Core Standards for Mathematics – Grades 9-12

• MP.2 – Reason abstractly and quantitatively
• MP.4 – Model with mathematics
• MP.6 – Attend to precision

• A-SSE.A.b:  Interpret complicated expressions by viewing one or more of their pairs as a single entity
• A-SSE.2:  Use the structure of an expression to rewrite it.

• A-CED.2 Create equations in two or more variables to represent relationships between quantities, graph equations on coordinate axes with labels and scales
• A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations

• F-IF.4:  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
• F-IF.5:  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

• F-BF.4.b:  Verify by composition that one function is the inverse of another.
• F-BF.5:  Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

• F-LE.5:  Interpret the parameters in a linear or exponential function in terms of a context.

• F-TF.1:  Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
• F-TF.2:  Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• F-TF.3:  Use special triangles to determine geometrically the values of sine, cosine, tangent and use the unit circle to express the values of sine, cosine, and tangent in terms of their values for x, where x is any real number.

 
 
C. Common Core Standards for English/Language Arts (ELA) – Grades 9-12

• RST.11-12.2 – Determine the central ideas or conclusions of a text; summarize complex concepts, processes, or information presented in a text by paraphrasing them in simpler but still accurate terms.
• RST.11-12.3 – Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text.

• RST.11-12.9 – Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.

• RST.11-12.10  By the end of grade 12, read and comprehend science/technical texts in the grades 11-CCR text complexity band independently and proficiently.