3. Vectors

Adding Vectors by Components

Vector Component Addition and Subtraction

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/631d351c39ab4d14-1676668315030.jpg" alt="" />

A position vector in the first quadrant has an x-component of 6 m and a magnitude of 10 m. What is the value of its y-component?

Adding Vector Components Example | Physics with Professor Matt Anderson | M3-11

Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward’s displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

A runner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. Let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. The runner starts at (x,y) = (50m, 0m) and runs 2.5 times around the track in aclockwise direction. What is his displacement vector? Give your answer as a magnitude and direction.

While vacationing in the mountains you do some hiking. In the morning, your displacement is Sₘₒᵣₙᵢₙ₉ = (2000 m, east) ＋ (3000 m, north) ＋(200 m, vertical). Continuing on after lunch, your displacement is Sₐբₜₑᵣₙₒₒₙ = (1500 m, west) ＋ (2000 m, north) ─ (300 m, vertical).

b. What is the magnitude of your net displacement for the day?

Vector Addition By Components

Vectors - Finding the components of a vector

Let <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mn>2.0</mn><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>, <math display="inline"><mrow><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mo>&#x2212;</mo><mn>7.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover></mrow></math>, and <math display="inline"><mrow><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mn>3.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>. What is the direction of <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover></mrow></math>? Express your answer in degrees counter clockwise from the positive x axis.

Let <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mn>2.0</mn><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>, <math display="inline"><mrow><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mo>&#x2212;</mo><mn>7.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover></mrow></math>, and <math display="inline"><mrow><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mn>3.0</mn><mover accent='true'><mi>i</mi><mo>&#x005E;</mo></mover><mo>+</mo><mover accent='true'><mi>j</mi><mo>&#x005E;</mo></mover></mrow></math>. What is the magnitude of <math display="inline"><mrow><mover accent='true'><mi>A</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>B</mi><mo>&#x21C0;</mo></mover><mo>+</mo><mover accent='true'><mi>C</mi><mo>&#x21C0;</mo></mover></mrow></math>?

Adding Vectors: How to Find the Resultant of Three or More Vectors

Anderson Video - Adding Vector Components Example

A velocity vector 40 degrees below the positive x-axis has a y-component of -10 m/s. What is the value of its x-component?

Adding Vectors

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium’s average velocity for the entire trip? <img src="https://lightcat-files.s3.amazonaws.com/problem_images/93e1dbb5a85bb398-1677783838193.jpg" alt="" />

A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components. <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/93083a9e0fc31bff-1678190867393.jpg" style="width: 300px;">

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (d) the vector difference B - A <img src="https://lightcat-files.s3.amazonaws.com/problem_images/c27460d9a4ad6484-1673442520803.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/1a1966ce78d4fae9-1678190776065.jpg" style="width: 182px;">

Anderson Video - Adding Vector Components

Anderson Video - Adding Vectors Example

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (a) the vector sum A + B <img src="https://lightcat-files.s3.amazonaws.com/problem_images/91ff45272b60d5fe-1673475545736.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/1468aa562fe00b6b-1678190695545.jpg" style="width: 178px;">

While vacationing in the mountains you do some hiking. In the morning, your displacement is Sₘₒᵣₙᵢₙ₉ = (2000 m, east) ＋ (3000 m, north) ＋(200 m, vertical). Continuing on after lunch, your displacement is Sₐբₜₑᵣₙₒₒₙ = (1500 m, west) ＋ (2000 m, north) ─ (300 m, vertical).

a. At the end of the hike, how much higher or lower are you compared to your starting point?

Physics Vector Addition (Algebraic)

Adding Vector Components | Physics with Professor Matt Anderson | M3-07

FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.
<img src="https://lightcat-files.s3.amazonaws.com/problem_images/9844d2ece039903b-1676567203420.jpg" alt="" />

Anderson Video - Displacement Vectors and Adding Vectors

For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (c) the vector difference A - B <img src="https://lightcat-files.s3.amazonaws.com/problem_images/78f96ca97e8dd6a9-1670246445278.jpg" alt="" /><img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/ee074203d82f1a52-1678190795461.jpg" style="width: 171px;">

Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

Patrick

Physics

https://static.studychannel.pearsonprd.tech/courses/physics/thumbnails/physics

#FFFFFF

Scott Teaches Physics

PhunScience

Physicshelp Canada

Patrick Ford

Physics with Professor Matt Anderson

Jennifer Cash

Professor Anderson

1. Intro to Physics Units

2. 1D Motion / Kinematics

4. 2D Kinematics

5. Projectile Motion

6. Intro to Forces (Dynamics)

7. Friction, Inclines, Systems

8. Centripetal Forces & Gravitation

9. Work & Energy

10. Conservation of Energy

11. Momentum & Impulse

12. Rotational Kinematics

13. Rotational Inertia & Energy

14. Torque & Rotational Dynamics

15. Rotational Equilibrium

16. Angular Momentum

17. Periodic Motion

18. Waves & Sound

19. Fluid Mechanics

20. Heat and Temperature

21. Kinetic Theory of Ideal Gases

22. The First Law of Thermodynamics

23. The Second Law of Thermodynamics

24. Electric Force & Field; Gauss' Law

25. Electric Potential

26. Capacitors & Dielectrics

27. Resistors & DC Circuits

28. Magnetic Fields and Forces

29. Sources of Magnetic Field

30. Induction and Inductance

31. Alternating Current

32. Electromagnetic Waves

33. Geometric Optics

34. Wave Optics

36. Special Relativity

[BOOK PHYS] !CLUTCH

Calculating Dot Product Using Components

Unit Vectors

Adding Vectors Graphically

Introduction to Dot Product (Scalar Product)

Intro to Cross Product (Vector Product)

Review of Vectors vs. Scalars

Trig Review

Introduction to Vectors

Vector Composition & Decomposition

Calculating Cross Product Using Components

General Chemistry

Jules

Study the key concepts of Physics with descriptive videos, questions, and more

General Biology

Jason

Financial Accounting

Brian

Dive deep into Psychology with videos, practice problems, and more.

Psychology

Microeconomics

The first in a two-course sequence introduces the basic structure and functions of the human body

Anatomy & Physiology

College Algebra covers algebraic topics and skills necessary to succeed in college-level math

College Algebra

Organic Chemistry

Johnny

Genetics

Kylia

GOB Chemistry

Microbiology

Cell Biology

Analytical Chemistry

Study topics incliding trogonometric functions, identities and equations, solutions of triangles, complex numbers, and other topics as time permits

Trigonometry

Learn the toughest concepts covered in Precalculus with step-by-step video tutorials and practice problems by world-class tutors

Precalculus

Biochemistry

Macroeconomics

Chemistry

Biology

Math

Business

Social Sciences

Michelle Barry Franco

Building Great Presentations

Eric Zackrison, PhD

Critical Thinking And Problem Solving

Sam Bennett

Embracing Times Of Uncertainty

Nicole Tetreault, PhD

Emotional Intelligence Basics

Eugene Choi, PharmD

Enhancing Your Critical Thinking Skills

Jason Richardson, PsyD

Habits To Win Every Day

Alyssa Lowery, PhD

Having Powerful Advanced Conversations

Don Gilman, EdD

How To Influence & Persuade

Amy Matthews

Leadership Skills For Rising Stars

Listening To Lead

Michael Shermer, PhD

Making Decisions Under Uncertainty

Lynne Hurdle

Mistakes To Avoid When Resolving Conflict

Michael Veltri

Overcoming Decision Making Traps

Mike Vardy

Prioritizing Effectively

Secrets Of Effective Prioritization

Strategies To Build Up Your Creativity

Tools For Decision Making

Tutti Taygerly

Unleash And Advance Your Creativity

Winning Collaboration: How To Work More Effectively With Different Personalities

Writing Emails People Want To Read

Critical Thinking

Self Management

Digital Literacy

Collaboration

Communication

Analytical Thinking

Leadership

Flexibility/Adaptability

Innovation

Eric Zackrison

Nicole Tetreault

Dive deep into Python with videos and more.

Introduction to Python

Arianne Dee

Agile Project and Product Management

Agile & Product Management

Mariya Breyther 

Introduction to Microsoft Power BI

Microsoft Power BI

Chris Sorensen

Data Analysis Fundamentals with Excel

Data Analysis - Excel

Chris Sorensen & Ammul Shergill

Getting Started with  and Cryptocurrency

Introduction to Blockchain

George Levy

Learn Enough HTML, CSS and Layout to be Dangerous

HTML, CSS & Layout

Lee Donahoe & Michael Hartl

Learn Enough JavaScript to be Dangerous: A Tutorial Introduction to Programming with JavaScript