Hello class, Professor Anderson here. Let's take a look at an example problem this is one that we already talked about but let's take a look at it again with this idea of breaking the vectors into components in terms of these unit vectors and then adding them up as vectors. Okay. So let's review our picture. We're gonna run 1 mile to the north, then 0.4 miles west and then 0. mile south. Okay. So our first leg of our journey is straight up one mile and then we go zero point four miles to the west and then we go zero point one miles south. And we want to figure out this guy right here. What is that R vector? Once you identify the vector you know everything about it. Okay? You know the length and you can calculate the angle. So all we need is the vector and let's label these A, B, and C, and let's figure out how to write these things in terms of the unit vectors that we just talked about. Okay? So this one, this first leg that's vector A. How do I write that as a vector in this XY coordinate space? I need something multiplying i-hat and then I need something multiplying j-hat. i-hat is in the X direction in this case it would be along the east axis. So, what do I put there? What should I put there? Yeah. >> (student speaking) Um. Zero. >> Zero. All right. This first one doesn't go any in the East. Okay. What do I need to put on the second term? One. Right? One mile. So this is how I write that first vector it's zero i-hat, plus 1 j-hat. The second vector, B, is gonna be what? Well we've got something in front of the i-hat. We have something in front of the j-hat. What do you think? Yeah. >> (student speaking) I'm confused as how- where did the zero come from. >> Okay, good question. So, this East axis is what we're gonna say is our typical x axis. Which means that i-hat points along that axis. Okay? So for the first vector, vector A, was it going at all in the i-hat direction? It was going exactly zero in the I hat direction. Okay. So we don't technically have to write that we just cross it out entirely but we're trying to be clear here. Alright, and of course it went 1 in the other direction, one mile in the other direction which is j-hat. What about B? Let's do the easy part first what should I put there on the j-hat for vector B? How far north is it going? Was that a okay or was that a zero? That's a zero. That's a zero, right? Vector B is going perfectly West it's not going at all north. What do we put on the first one? You're itching to raise your hand and shout something out, go for it. >> (student speaking) It should be 0.45. I don't know if it's negative. Is it supposed to be negative? >> All right. Let's do the easy part first 0.4. And now the question is it positive or is it negative? Alright. When you are moving along the x axis, its positive. If I'm moving along the negative x axis, then it is negative. So is that thing moving to the right or to the left? >> (student speaking) To the left. It's moving to the left, and so we want to make this negative. Good. All right, almost there. Finally we got to do vector C. What is vector C? Well we know that it's not moving in the east or west, so that first term is zero. And now based on what we just did, what should I put there? You guys know the answer to this it's negative 0. Okay? All right. Those are our three vectors written out in this component form and now it's nearly trivial to write the resultant vector. How do I do it? I add these up. All the i-hats I add up. I add to negative 0.4 to 0 and of course I get negative 0.4. And now I add up these 3. I have 1 plus 0 plus negative 0.1. And that gives me a positive 0.9 j-hat. And now you're essentially done. Once you know the vector you essentially know everything about the vector. Right? Calculating the magnitude is just a matter of squaring the first term, squaring the second term, and taking the square root and we already did this we know what the answer is it's 0.98. And if you want to figure out say this angle right here, phi, how do we do that? Well let's draw the triangle phi this is 0.98 which we said was R. This was 0.9 which that's going to be our parse of Y and this is our R sub X which was negative 0.4. Okay. Those are the three sides of our triangle this is again a right angle and so phi you can do any trig relation you like, but let's take the arc cosine of 0.9 over 0.98, and if I remember right I think we got 23 degrees. Okay? Any questions about this one? Any questions about how we approached it with the unit vectors? Okay, good. Hopefully that one is clear, if not come see me an office hours. Cheers.
Table of contents
- 0. Math Review31m
- 1. Intro to Physics Units1h 23m
- 2. 1D Motion / Kinematics3h 56m
- Vectors, Scalars, & Displacement13m
- Average Velocity32m
- Intro to Acceleration7m
- Position-Time Graphs & Velocity26m
- Conceptual Problems with Position-Time Graphs22m
- Velocity-Time Graphs & Acceleration5m
- Calculating Displacement from Velocity-Time Graphs15m
- Conceptual Problems with Velocity-Time Graphs10m
- Calculating Change in Velocity from Acceleration-Time Graphs10m
- Graphing Position, Velocity, and Acceleration Graphs11m
- Kinematics Equations37m
- Vertical Motion and Free Fall19m
- Catch/Overtake Problems23m
- 3. Vectors2h 43m
- Review of Vectors vs. Scalars1m
- Introduction to Vectors7m
- Adding Vectors Graphically22m
- Vector Composition & Decomposition11m
- Adding Vectors by Components13m
- Trig Review24m
- Unit Vectors15m
- Introduction to Dot Product (Scalar Product)12m
- Calculating Dot Product Using Components12m
- Intro to Cross Product (Vector Product)23m
- Calculating Cross Product Using Components17m
- 4. 2D Kinematics1h 42m
- 5. Projectile Motion3h 6m
- 6. Intro to Forces (Dynamics)3h 22m
- 7. Friction, Inclines, Systems2h 44m
- 8. Centripetal Forces & Gravitation7h 26m
- Uniform Circular Motion7m
- Period and Frequency in Uniform Circular Motion20m
- Centripetal Forces15m
- Vertical Centripetal Forces10m
- Flat Curves9m
- Banked Curves10m
- Newton's Law of Gravity30m
- Gravitational Forces in 2D25m
- Acceleration Due to Gravity13m
- Satellite Motion: Intro5m
- Satellite Motion: Speed & Period35m
- Geosynchronous Orbits15m
- Overview of Kepler's Laws5m
- Kepler's First Law11m
- Kepler's Third Law16m
- Kepler's Third Law for Elliptical Orbits15m
- Gravitational Potential Energy21m
- Gravitational Potential Energy for Systems of Masses17m
- Escape Velocity21m
- Energy of Circular Orbits23m
- Energy of Elliptical Orbits36m
- Black Holes16m
- Gravitational Force Inside the Earth13m
- Mass Distribution with Calculus45m
- 9. Work & Energy1h 59m
- 10. Conservation of Energy2h 54m
- Intro to Energy Types3m
- Gravitational Potential Energy10m
- Intro to Conservation of Energy32m
- Energy with Non-Conservative Forces20m
- Springs & Elastic Potential Energy19m
- Solving Projectile Motion Using Energy13m
- Motion Along Curved Paths4m
- Rollercoaster Problems13m
- Pendulum Problems13m
- Energy in Connected Objects (Systems)24m
- Force & Potential Energy18m
- 11. Momentum & Impulse3h 40m
- Intro to Momentum11m
- Intro to Impulse14m
- Impulse with Variable Forces12m
- Intro to Conservation of Momentum17m
- Push-Away Problems19m
- Types of Collisions4m
- Completely Inelastic Collisions28m
- Adding Mass to a Moving System8m
- Collisions & Motion (Momentum & Energy)26m
- Ballistic Pendulum14m
- Collisions with Springs13m
- Elastic Collisions24m
- How to Identify the Type of Collision9m
- Intro to Center of Mass15m
- 12. Rotational Kinematics2h 59m
- 13. Rotational Inertia & Energy7h 4m
- More Conservation of Energy Problems54m
- Conservation of Energy in Rolling Motion45m
- Parallel Axis Theorem13m
- Intro to Moment of Inertia28m
- Moment of Inertia via Integration18m
- Moment of Inertia of Systems23m
- Moment of Inertia & Mass Distribution10m
- Intro to Rotational Kinetic Energy16m
- Energy of Rolling Motion18m
- Types of Motion & Energy24m
- Conservation of Energy with Rotation35m
- Torque with Kinematic Equations56m
- Rotational Dynamics with Two Motions50m
- Rotational Dynamics of Rolling Motion27m
- 14. Torque & Rotational Dynamics2h 5m
- 15. Rotational Equilibrium3h 39m
- 16. Angular Momentum3h 6m
- Opening/Closing Arms on Rotating Stool18m
- Conservation of Angular Momentum46m
- Angular Momentum & Newton's Second Law10m
- Intro to Angular Collisions15m
- Jumping Into/Out of Moving Disc23m
- Spinning on String of Variable Length20m
- Angular Collisions with Linear Motion8m
- Intro to Angular Momentum15m
- Angular Momentum of a Point Mass21m
- Angular Momentum of Objects in Linear Motion7m
- 17. Periodic Motion2h 9m
- 18. Waves & Sound3h 40m
- Intro to Waves11m
- Velocity of Transverse Waves21m
- Velocity of Longitudinal Waves11m
- Wave Functions31m
- Phase Constant14m
- Average Power of Waves on Strings10m
- Wave Intensity19m
- Sound Intensity13m
- Wave Interference8m
- Superposition of Wave Functions3m
- Standing Waves30m
- Standing Wave Functions14m
- Standing Sound Waves12m
- Beats8m
- The Doppler Effect7m
- 19. Fluid Mechanics2h 27m
- 20. Heat and Temperature3h 7m
- Temperature16m
- Linear Thermal Expansion14m
- Volume Thermal Expansion14m
- Moles and Avogadro's Number14m
- Specific Heat & Temperature Changes12m
- Latent Heat & Phase Changes16m
- Intro to Calorimetry21m
- Calorimetry with Temperature and Phase Changes15m
- Advanced Calorimetry: Equilibrium Temperature with Phase Changes9m
- Phase Diagrams, Triple Points and Critical Points6m
- Heat Transfer44m
- 21. Kinetic Theory of Ideal Gases1h 50m
- 22. The First Law of Thermodynamics1h 26m
- 23. The Second Law of Thermodynamics3h 11m
- 24. Electric Force & Field; Gauss' Law3h 42m
- 25. Electric Potential1h 51m
- 26. Capacitors & Dielectrics2h 2m
- 27. Resistors & DC Circuits3h 8m
- 28. Magnetic Fields and Forces2h 23m
- 29. Sources of Magnetic Field2h 30m
- Magnetic Field Produced by Moving Charges10m
- Magnetic Field Produced by Straight Currents27m
- Magnetic Force Between Parallel Currents12m
- Magnetic Force Between Two Moving Charges9m
- Magnetic Field Produced by Loops and Solenoids42m
- Toroidal Solenoids aka Toroids12m
- Biot-Savart Law (Calculus)18m
- Ampere's Law (Calculus)17m
- 30. Induction and Inductance3h 37m
- 31. Alternating Current2h 37m
- Alternating Voltages and Currents18m
- RMS Current and Voltage9m
- Phasors20m
- Resistors in AC Circuits9m
- Phasors for Resistors7m
- Capacitors in AC Circuits16m
- Phasors for Capacitors8m
- Inductors in AC Circuits13m
- Phasors for Inductors7m
- Impedance in AC Circuits18m
- Series LRC Circuits11m
- Resonance in Series LRC Circuits10m
- Power in AC Circuits5m
- 32. Electromagnetic Waves2h 14m
- 33. Geometric Optics2h 57m
- 34. Wave Optics1h 15m
- 35. Special Relativity2h 10m
3. Vectors
Adding Vectors by Components
Video duration:
7mPlay a video:
Related Videos
Related Practice