Vectors for Physics (Mathematics for High School Physics, part 2) | Edouard RENY | Skillshare

# Vectors for Physics (Mathematics for High School Physics, part 2)

#### Edouard RENY, Music Producer & Tutor in Physics

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• ### 5. Vectors: Training Exercises

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## About This Class

In High School Physics, maths is just a tool… like a hammer or a screwdriver. This course shows you how to use such a tool, in the perspective of a physicist, meaning a practical perspective.

This course contains three sections divided in three classes:

1 - Algebra for Physics

2 - Vectors for Physics (the class you are consulting now)

3 - Trigonometry for Physics

This class “Vectors for Physics”, teaches all that is required to know about Vectors in regards to solving Physics question.

Section 2 – Episode 1: What is a Vector?

The first video discusses what a vector is. Then, it presents two ways of describing it using either Cartesian coordinates or Polar coordinates. Finally, this episode teaches how to convert a set of vector coordinates from one type to the other. This operation is incredibly common and useful in Physics.

Section 2 – Episode 2: Adding Vectors

Knowing how to add vectors to one another is crucial to any students in Physics. This episode will show how to do so graphically and algebraically. It contains formal lessons and solved examples.

Section 2 – Episode 3: Solving Physics Problems with Vectors (in 1D and 2D)

The third video shows how the maths about vectors presented in episode 1 and 2 come together. It teaches how to solve Physics problems with vectors in 1 dimension, and in 2 dimensions.

Section 2 – Episode 4: Vectors – Training Exercises

The final video of this section is composed of two full blown exercises that involve vectors. The first exercise deals with electric charges, and the second one, with gravitational forces. Actually, the student is required to use his/ hers understanding of vectors to save a spaceship lost in an asteroid field! These questions can be solved without any prior knowledge of electrical and gravitational fields.

## Meet Your Teacher

#### Edouard RENY

Music Producer & Tutor in Physics

Teacher

Edouard Andre Reny was born in 1971 in Bordeaux, France. Long studies in sciences armed him a PhD in solid state chemistry which led him to a post doctorate contract at Hiroshima University, Japan. In his early thirties, he integrated a large water treatment corporation in The Netherlands as a senior researcher. A decade later, he decided to fly with his own wings by founding his own company, “Synaptic Machines”, that brought together his interests in sciences, his drive to share it with the world, and his passion and talent for music. Why not make a living with what one truly loves!

This coincided with the realisation that he was a damn good teacher. To support financially his bran new company, he started tutoring a few kids in their late teens to prepare for their I... See full profile

## Related Skills

Technology Mathematics Physics Data Science Vectors Trigonometry

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1. Vectors for Physics: you are in No. Two last years of high school and you took physics. But it is so hard, it doesn't have to be that way. Physics is actually quite easy. And once you realize this, you will also realize how this subject is fascinating. Many of my students get stuck on. They have trouble applying when they learned math class to physics. This is why I created this course, a bridge, a bridge between math and physics. In this course I review with you all the math. You need to feel comfortable in your study of physics. At high school level, there are three sections. A dribble. Well, you will learn how to rearrange physics equations effectively and reliably vectors so important in physics off which the manipulation you will learn to master. And finally, we will dive into trigonometry and its application to vectors. As about this dispersed in the course. They are also talks of little practical tips to make life as a physics student easier, follow their schools with a pen, paper the calculator and be ready to pull the videos. Lessons are punctuated with solved examples and questions for you to train. They are also videos that compile exercises, including exam questions for you to work and in these videos, a solve these questions in grand detail on a white board. Take this course followed diligently. Work on all the exercises and math will never be a problem again Studying. 2. What is a Vector?: victims. That is our essential mathematical tool that you need to master in order to enjoy your physics course. In this video, I will introduce you to Victor's what they are and how to use them. So without further ado, let's get started. Let's imagine a town a send him to draw the town a like a little house next to Tom A. This town be in town a. There's John and John had a question. He wonders where can be is located. So I talked to him and says, Oh, hi, Emma West town be in our answers. Oh, Tommy is not far is just five kilometers away. What information did John get? If you drool town A here. Well, tell me would be five kilometers away. So it could be, for example, there. This will be five committed. It could also be that basically, town, we will be located on a circle centred allowed town A of radius, five kilometers. So if John has lots of courage, well, he will just walk five kilometers away from town a and then describe a circle around it until it forms. Tell me that's nothing effective. So he realizes that on he asked them, I'll get Yeah, but could you be a bit more precise? Please? Halima says. Oh, sure, Tom, these five kilometers away in the Northeast direction up. So we have more information Now. We know that Town B is located here in the Northeast. Damage so Northeast. Compared to what? Yeah, it's a direction, but you have to have a restaurant to have a direction. Well, the restaurants is within the name itself because it's in reference to the north and to the east. So you have the nexus here, which is the north and nexus here, which represent east and Northeast. We know about convention that northeast is in between, so we have a length five winter and we have a direction now. The five kilometer is just number. It's called a scallop. It's a quantity with suggested number Here. What we can call a B is a vector because it has a magnitude, a link and a direction. So we could represent maybe that way 80 in top of a B to express it. It is a vector. I will put a little arrow to show it has a direction, is five kilometers long and in the direction off northeast northeast is not a precise Yeah , because you got north, east, north, east southeast etcetera who had basically a directions for full circle. So it's not precise enough. You can improved the precision by saying OK, north, east, east. So a northeast east direction would be in the direction which is in between northeast and east. But that's about it. Just as eight extreme directions. You want to use something more precise, you will use angles. Yes, you have 360 degrees within a circle, so it's not precise. It's number, so you can have also a decision. So Northeast is actually 45 degrees. I be five kilometers, 45 degrees, good, 45 degrees and weapons to what to the X axis on the positive direction would be this way. We will take this way. It gives you positive values that's assumed right. So you have to coordinates within. Be a projector five kilometers, which is a late let's go, the magnitude. You're right. It may be like this without putting. I will all You can also write it like this. And that's of course to five kilometers. That's called the magnitude of the vector. The 45 degrees represent the direction of the vector for the angle of director direction, Direction and magnitude are cord innings. These are the coordinates of the vector. Why? Because if you give a magnitude like in the direction you describe a vector in a unique way , they cannot be another vector, which has the same Cordy. Next these are called polar coordinates. On my board is not big enough. Oh, yes, it is just so I represented our town a and tell me on what describes a position will be related to a It's a vector a B off magnitude five and off angle compared to the x axis for Fanta, please. So these are polar coordinates. There's also another way to represent vector using unicorn innings Kardashian Corden eggs. Let me explain. I'm drawing a reference fame, X y and acknowledging. Oh, and I take your points And how do you describe I m in a unique way? Well, I describe end by it's called me. These are called Cartesian coordinates, but at the same time, I'm describing in Vector o m. Yes, The Victor O. M. Has the same coordinates as a point en because O is theology and what's even more interesting is that X and Y themselves elected because I have a direction. Yeah, you go from there to there, that's X. And then you add this vector Why two x to get away So you got a ran equals X plus life. X and Y are called Cartesian coordinates off Victor and they also called the components off the vector. Remember this term component? Ex and why are the components of the vector of them and om? It's a resultant off the vector x and y When you have X plus y, it results in him and what composes om the components x and y remember these terms that they usefully fizzy 90. Creating vectors in physics is very common. One off the operations you will need to do is to go from Poland Cartesian coordinates and from Cartesian to Pola. So let's go from Poland. Kardashian, I've got the vector a B, which has a polar coordinates its A B, which is a number and data, and I want to find X and what you recognize here, maybe a rectangle triangle. Let's do it. Be okay, Get up on this is the value, baby. Well, X will be this side. And why will be that one? You know that cost data is adjustment of hypothermia. The adjustment high profits, you know, case it will be x of a B giving ex equals cost. Also a b. Acosta. And you know that scientists are yes, composite overhype Athenians. That is why of a B. So why equals a B sign? So if you have okay, angle on the magnitude of vector, you can find its Cartesian coordinates its components to symbolize when you're transforming polar coordinates into Cartesian coordinates. It means that you are taking the magnitude and the direction off the victor. And from that you're calculating its components to finally always mental component. You just multiply the magnitude by the costs off the angle. And to find a vertical component, you must apply the magnitude by the sine of the angle on. Be careful. This angle is the angle in reference to the X axis. So practically what does it mean? Let's go back to physics and see a practical case. This case is under the form of an exercise. The plane takes off with an angle off 15 degrees and its velocity is 100 feet, deeming just per second calculate. It's always until and vertical velocities. So maybe you want to give it a try fist. So pause the video and I'll be back in a few seconds for the correction I represented on the board the velocity vector of the plane here. So we have the magnitude. I don't have the direction in reference to the or even told accents. The question is to find the components of the sector because we're looking for me. Always enter velocity on the vertical velocity is on the compartments off the velocity vector. So you're looking for these RDX on view? Why? Well, it's simple. Tree VX is equal to the magnitude of the vector, so 150 multiplied by the cost Sign off the angle Cost 15 on the y component. Vertical velocity is 150 and to plan by sign 15. So let me use my calculator out of 15 by costs. 15 gives me on 14. Five me just a second on 100 50. Supplied by sign off 15 39%. The villa. They may be going from polar coordinates to Kardashian. Cool evenings just means that I'm looking for the components of Victor one off. The most common manipulations with Victor's in physics is to transform Cartesian coordinates into pole. According so suppose that time a vector A B and I know the coordinates X and y What I'm looking for here is get up on the magnitude A B. So I'm transforming a B. They're not transforming. I'm finding the new coordinates. Or maybe from Kardashian two polar. I'm looking for a B and C here, X and y I know. So it's gonna be easier for you. Maybe if I give you some numbers. Suppose that I have a vector a B X equals two and why would speak? And then I'm still looking for the magnitude now. Maybe so how have electing strangle you. So I know that a B squared equals X squared class wise quit. Therefore I can find out a b which is dust quivered off X squared. That's why it's quit on. That's gonna be equal to full plus nine equals. Quote 13. The nine inches off my vector will be squared of 13. Now that we have the magnitude, let's look for the angle to do that, I can Just remember that's X was equal to the magnitude by Kosta and why was equal to the magnitude by scientists. If I lie down, why over eggs? I get a B scientist toe of a be costed giving me scientists are over cost. It are because every goes away tank with them. So titter is tangent minus one. Why of X so to find polar coordinates from Cartesian coordinates. We need these to formula to symbolize Transforming Cartesian coordinates into polar coordinates means that you are using the components off the vector to find its magnitude and its direction. Mine into the other direction defines a vector so often in a physics problem, you will have the components X and Y And they were asking the question Okay, what is the force say so from the components of the force the X component of the Y component. You can calculate the magnitude of the force and then what direction it's going. You can do that by applying formulas. Magnitude of a vector is equal to the square root off the sum of the squares of its components and the direction of the vector defined by the angle is equal to tension minus one off the ratio between the Y and X component. Note that the X component is on the denominator. That means that the angle is in reference to the X axis for you to get a feel of what it means physically, let's work on a little exercise. A boat is sailing east at 30 kilometers per hour. At the same time, a strong northern wind makes its light north at 10 kilometers. After 10 hours, what distance will have the boat covered and in what direction? If you want to try this exercise, pause the video and I'll be back in a few seconds. We have the components of the velocity. That means we know its Cartesian coordinates. By transforming these into polar coordinates, we can find the magnitude and direction off the velocity. Then we can use the magnitude of the velocity, which is also the speed to find the distance covered by the boat. After 10 hours, V equals distance of the time. The magnitude of velocity is equal to the square it off the some off the squares of its components. So be X squared. Bless, view wise, quit that is 30 squared plus 10 squared, and it was quoted so 30 squared is 900 10. Squared is 100. So the magnitude of the velocity is square root of 1000 and that is 31.6 kilometers. Wow, that's a speed of the boat, the magnitude of the velocity so we can find the distance right, because the speed is distance covered by the time so distance is V by time. The 31.6 kilometers through our town is also in hours, 10 hours, so we can use 10 giving, feeling 16 kilometers. So after 10 hours, the boat will be 316 kilometers away from this point. But in one direction, 316 that way, that way, that way, Well but entity leave. You know it's gonna be around here, so let's find out the angle so I'll boat is somewhere there. Let's find out the angle. We can find it out by using the Formula 10 minus one off the Y component over the X component. So that's 10 minus one off 10 divided by 30 and that is 80.4 degrees. So the direction is 18.4 degrees in reference to the X axis. In this episode, we have understood that vectors are quantity contained both magnitude and direction. We also realize that we can describe vectors in two different quoting eight sisters, Cartesian coordinates and polar coordinates. Finally, we learned how to convert the coordinates of a vector for one system to the other, an operation which is very common in physics. Actually, all the other videos off this section show how such an operation is essential to master. In the next video, you will learn how to add vectors graphically and algae brightly. The third appears old. We'll show you how such knowledge is applicant doing physics problems. By then, you should have become a victim master, and it would be time to train your new skills. In the last episode of this section, fish contains practical exercises 3. Adding Vectors: vectors are mathematical concept, it appears a little everywhere in physics. Velocity, displacement, acceleration forces, magnetic flux, density, gravitational shear, strength, electrical fear, strength, aria momentum. So many physics para meters are vectors, so it is essential to get a grasp of what vectors are and how to manipulate them. In the first video about Victor's, we learned that the vectors a quantity that contains both a magnitude and the direction, for example, the force on the boat. It has both a magnitude and the direction. We also realized that the vector could be described uniquely in space by two different systems of co ordinates. Polar coordinates describe vector with this magnitude and election in reference to one access. So you have the magnitude which is the length of the victor, and you have the direction which is described by the angle in reference to the access. Doctors can also be described by Cartesian coordinates in reference to two Purple nickel access. X represents how much the vector extends in the office rental direction and why how much it extends in the vertical direction. This implies that the coordinate of the vector are not linked to the origin point or any point. As you can see in the graphic In the previous video, we learned how to convert from polar coordinates to Cartesian coordinates. This means finding the office gentle and vertical components off the vector for mix magnitude and its angle in reference to an access. We also learned how to calculate the magnitude and orientation of the vector. For me, it's always entitled, and vertical components this course stones to converting Cartesian coordinates into polar coordinates. X. In this video, we we learned something really important in physics. How to add vectors to one another. We we learn how to do this graphically, and also after brightly, we have a vector A and the Vector B. We want to find out what the vector a Class B looks like. So for that, let's do Victor A again. Then from the tip off that, to a drawback, to be the vector A plus B will start on the tail off Victor A and end at the tip off. Victor be You know that this basement is a vector white. When you can visualize a man walking along a then along B and the total change of position , he will have experienced is the one described by the vector A plus B. What about subtracting back to be from Victor A. This means finding a victor a minus bi. To be truthful. Nobody knows how to do that, but what we know is that you can add Vector A to a vector minus B. So let's draw back to a on from the tip of Victor A. Let's draw victor minus B. Dr Minus T has the same magnitude and orientation as vector plus B, but in the opposite direction. So the victim a minus bi. We'll start on the tail of Victor A. An end at the tip off Victor minus B. To illustrate this, let's do a quick exercise. We have three vectors X y and zed, which are vector A, B, C or D, corresponds to X plus y minus, said Force a video and figure it out. First, let's pro Vector X. Then from the tip of the X, let's start victor. Why? The next operation is a subtraction. So let's add minus said, starting from the tip off White. Now the vector joining the tale of X and the tip off minus said, will be the solution. Let's review the answers proposed in the exercise. Yes, the answer is vector B. Now that you know how to add vectors graphically, let's do another exercise. A car is making a circle return at Causton. Speed the velocities off the card Point em and end I represented by the red vectors. Which of the following vectors were present? The change in velocity of the car from point M two point n. We are looking for a change in velocity Delta v. Delta V, which is the n minus of the end, so we can translate this in the N plus minus B. Let's draw this. We first drove the end. So that's the end and we draw miners. The end sits viene but flicked. That's minus the end, says the change of velocity. We stop here and then is there that sounded heavy. Let's check the exercise. It's also see we just learned how to add vectors. Graphically, geometrically. Let's learn how to add them. How did you break it? So I presented two vectors A and B, and we know the magnitude and the angle between them. But for simplicity, I'm going to move, be in order to make it start at the tip off. A. That's not going to be first step. They find some access they find to access, which are perpendicular. I recommend that you choose the access carefully choose them so that they're parallel to the maximum number of vectors, which are involved in a problem. So in our case, I would choose the next access along. A when they speak the positive direction and why access need to be perpendicular, so I choose by away access. By best the angle. He'll is 1 10 but this looks more like 1 20 so I'm just going to change it. That's cool. Having a white board second step, take all the X components off the vectors and add them together. This will give me the X component off the son Victor. Let's define our some vector like being see so I can do it graphically. That black that would be see So we have C equals a plus B. We're going to try to find the magnitude of C as well as its orientation compared to X I this. So let's take the some off all the X components, so that would be c X What would be the X component off? A. Well, it's just okay, so I'll just write it like this. A X plus B X. So that's a plus. Be costs off this angle. We know this angle is 1 20 So this anguish 60 degrees. So for 60 let's keep some values to A and B uh, they could be 20 on B could be 10 so that would be equal to course of 60 is 1/2 B is 10. So that's five to which had 20 25. Let's do the same thing for why we're still in the second step there. We finding the components off the some vector. So see why Eventually stop the video, find it yourself, pulls a video. So what did you find? The y component off a. Well zero. Because A is perpendicular to the UAE, Access the white components of B will be the projection of B on this axis on the way Accents. What's the angle here? 30 degrees So I can do be cost 30. What I prefer be signed 60. So this sign I think so be it is 10. Sign 60 is open 866 That gives me 8.66. I have the Cartesian coordinates off Victor See, so the final step is quite simple. I just supply the formula to go from Kardashian to Pola, so step number three see the magnitude of Vector C is equal to squared off square of 25. Thus square off 8.66. That gives me I finds quote of 700 which is 26.5 for the magnitude off the vector sum. See, let's look at the angle. Let's look at the donation. Did that It would be this. Take the Anglo versus the exact sense you can use the formula. Tangin minus one off the Y component 8.66 divided by the x component. Five. Actually, this I find 1921 degrees. This check of this makes sense. Competitive graphic. So is 20. See you guys in the thirties. Let me Strong twenties. Yeah, that makes sense With the angle 20 degrees makes sense to. So let's some allies. What do you want? To add two vectors. First thing you do is actually to redraw the vectors in order for them to touch. That's make things easier to see. Then Step one. They find two perpendicular access. Choose them carefully in order to minimize calculation. Step number two. Find the components of the Vector song. You do this by adding the components off each individual vector along one direction and then step three. Once you've got the components for the vector sum ie the Kardashian coordinates, you just convert them to polar coordinates. I knew them. Congratulations. Now you know how to add vectors graphically and algae Brackley. But you might be wondering how to apply these techniques in the context of a physics exercise. So let me build a bridge between math and physics here, often in problems involving forces you will need to. Some, the force is applied on an object in order to get the result in force. Also called the Net Force forces are victors. So we need to apply the techniques we learned in order to add vectors to one another in the next episode. We we look at this in one dimension and in two dimensions, so make sure you check it out 4. Solving Physics Questions with Vectors (in 1D and 2D): In the two previous videos, we learned what vectors were how to convert their coordinates from Poland to Kardashian and visa versa. We also learned how to add the graphically and algebraic. Let's now see how all these bits of knowledge come together in solving problems in physics . We will start by looking at the situation in one dimension and then I'm great to two dimensions. After reviewing this video carefully, you will be able to succeed in solving. The exercise is presented in the next video. Imagine you have a great which is five kilograms on which is addressed, meaning that it is not moving. So draw the great five kilograms. It's no moving. Now you start pushing it. That is. You apply force on it off. 50 Newtons. What happens to the crate? Well, the great was stopped moving. You can imagine this in your life. If you put something addressed, it stops moving Starts moving Meaning that it was at velocity zero and now it has a certain velocities Has a velocity changed when you apply the force? If you have a change in velocity, that means you have an acceleration. So the force causes an acceleration That's the second Law of Newtown F equals M. A. We're going to this in more detail in another course so we can calculate the acceleration on great, which is F divided by so 15 delight by five. The acceleration is sweet meters, the second squared great, but usually the surface on which is create is not fiction list. So you can imagine that this great it's subjected to fiction. All right, so let's consider the fixture being say five new times. So how do you calculate the acceleration off the quake? Well, I need to find out how much force in total days on the creek. So there will be 50 noodles. Two works, five of the 50 Newtons. I going to be compensated by the fiction minus five. Give me 10 new times. So that would be my net force. Yes, so now I can apply new to Los F Net equals the May. So a equals 10 divided by five a equals two meters per second squared course. There's less effect now because the net force is only 10 meters. What did we do here? We added two forces. Let's do it graphically. Now I have a Victor F off 50 new terms. The lengths off the Arrow represents the magnitude of the victor, and to this I add a vector little F five mutants. But the other way. So actually I substructure did. And the length off my fiction is also represented. Table its magnitude five. So here we have 15 we have five. So the storm of these two things let me find a pen. We'll be starting from that point ending at that point. So this that would be my hopes. That would be my net force. And if I took a ruler and measured the length, I would find 10 unions. That's graphically start from there you got there. So at the end of the first vector, then from this point you put the other vector. But here it's 95 Newtons and then go on. That's starting point to end point and you can fill up the gap on that would be in that force with some of them to Victor's. Good. What about you Blankly? Well, we wanted to find the some of the forces because this is what F net is. The net force is the some off the forces so we have my net force is equal to f applied plus a fiction. Then I define an axis a positive access, so that would be my positive access. When I go to magnitudes, I can write down if nets equals 15 minus five because of five, Newtons is only getting sites of minus five giving me an F nets off 10 Utahns. So that was quite straightforward. You have a graphical solution and we have the algebra of solutions that this was in one dimension many times in physics, he will have to add vectors in order to solve problems. For example, here I devised one where you have all which is attached to a kite through a book. So the kite is feeling the wind. So is pulling the bull upwards my attention within the look. There's also the weight of the ball pulling the ball down. These are two forces with shocking enough opposing each other. The angle of the tension with horizontal direction is 60 degrees. So the question is to find the net falls because I want to know where the ball will be later. What is the direction the bull is taking? Is it going up, was it actually for? So we need to find the net force. It's not only do they in this direction, this is where we will add vectors. We will add the forces together to find this net force, and that's where we can apply our free step technique First. Choosy axis conveniently Second, find the components of the summer forces or some of vectors. And third is components being also the Kardashian cooling needs transform them into polar coordinates in order to get the magnitude on the direction off. The result of victor First step defined convenient access. So, for example, I could use this one for the X axis and this one upwards positive for the Y axis. Why did the truth these access? Because they contain a vector. I could have chosen one like this for X and one like this for why it's fine to. But that's easier to read, especially that I have the angle here with the always into elections. That was the first step. Second step. Find the components off the sum of vectors on each access. The component on this axis off the Victor song will be the some of the components off each individual vector. Let's look at the X axis. The component of the net force on the X axis will be, but we'll start at one point and we your around and you check all the vectors. Which involved? Does he have a component on the X axis? Yes. You can see it here. This is a component on the X axis of tea. What is its value? Well, the angle between the axis and the vector 60 degrees on the angle is between the victor and the axis sets. Two costs 60. That's continually to tour and check other vectors. Wait, Does wait have a component on the X axis? No, because you see, if you project the weight on the X axis, you just have a point zero wait is perpendicular to the XXY said no component on the X axis . You can put numbers. Yeah, let's define magnitude similar to what I drew. So 40 I would say 10 Utahns for weight, N g. I would put 20 new turns on We have the angle theta Okay, so I can remove the zero. You saw my reflects to track and remove it because it's just taking space. But let's look at the Y axis pulls a video. Try to find by yourself what I should like you. Let's start our little tour here like this. Check out all the forces and find out the white components of these forces. Tension. That's tension. Have a wife tones. Yes, it does. You can see it here. Project tension on Why? Well, you find this month the angle. There's no in between the vector and the access. I want to project the vector so it's sign. T sign 60. What about weight? Does it have a component? Yes, wait is parallel to the axis. Therefore, it's actually a full component. The magnitude off this component will be entry. But here the donation is needed. So it's gonna be minus injured. Let's begin numbers. T is 10. Cost sixties, one heart. So here's straightforward five Newtons T is 10 some 60 0.866 minus 20. So let me calculate that I find minus 11.3 noodles. That's it for Step two. We had the components off our net force, so we also have its Cartesian coordinates. We just need to convert them to pull a coordinates in order to get magnitude and direction . That is step three. So let's do this. One that was doing the access to that was finding components. Three will be to find magnitude and direction by converting Cartesian coordinates to polar coordinates. So the magnitude off the net force with equal to square root off the sum of the squares of opponents tries grad class minus 11.3 squared 12 point for new times. For the direction we can define an angle in reference to the X axis by using formula to find the Sango, which is 10 minus one off the Y component divided by the X component. So you guys 10 minus one off a minus 11.3, divided by five. Well, I see you've got a negative value here. That means I'm going to get and they get the angle. Remember the convention when you're going out it clockwise, it's positive. Clockwise is negative on this in reference to the X axis. So if I apply these numbers on my calculator, I find minus 66 degrees. These other two results magnitude and direction. Do these numbers make sense for that? We can check it by applying a graphical additional victors. What should I do this on the ball down. Do a little space here. Good. Let energy again and we'll need these members. I just need a space. I'm going to draw a team. T was something like this. And then she was downwards twice the size. So the result of vector would be so I've been right. Yeah. The result Nectar would be This doesn't make sense compared to the x axis. Yeah, about mine. 60 degrees In terms of size, A little bigger. The intention intention was 10. So 12. Yeah, that makes sense. Another way you can do this is by looking at the components off the Net Force. I don't know how green is going to shop on the screen because I'm using a green screen. I can draw the components. I know that F Nettle Banks. It's five. Muto's is actually the component of tea. F net of y is minus 11.3 2nd draw, minus living 0.3 things like this. And you see, if I argue to vector together, I get the same one. So it does make this kind of operations with vectors is very common in physics. So I recommend you get family with these manipulations to help you with that and train your new skills. The next video contains a couple of exercises in which we'll have to determine the summer off forces. The 1st 1 will involve electric charges repelling each other. The second exercise would be more intense. Hundreds of lives depend on new capacity to find the result of gravitational force on the station. 5. Vectors: Training Exercises: In the three previous videos about vectors, you have learned what a vector was and that it could be described within two different coordinate systems. Cartesian and polar. We also learned how to convert the coordinate of a vector from one system to another, for example, from Cartesian to Pola in order to get its magnitude and direction. And finally, you have learned how to add vectors graphically and agile, brightly. So now you are fully ready. And let's play with vectors. In this video, we would work on two exercises. The 1st 1 is a guided question that involves charges repelling each other in the 2nd 1 will have to take later net force and save a spaceship from destruction. So let's get started. A, B and C are all positive charges off. 10 micro cooler. They are set so that a B is perpendicular to BC. The charges repair each other. The force off charge A on Charles B is 0.9 new terms, the fourth off charge. See on Charles B 0.225 year terms. The first question you are required to draw a free body diagram off the forces on B tried to make it in scale because in the second question, you will have to estimate graphically the magnitude and the direction off the total force exerted on charge. Be by charges A and C. In the third question, you will be able to verify your graphical estimation because you will have to calculate the total force exerted on chance be using an algebraic method. Good luck. All charters are positive, so they have the same sign and the repair. The penalty means that they apply a force in each other, which is in the opposite direction. For example, a will repel be supply of force on me that way. See, we're repel be so apply a force of being that way that goes funds to drawing the people. Battaglia the force off A on B with the downloads. If off a on B, the force off see on me would be that way, but it's much smaller. We want to draw them at scale as it is required in the question. The force off see of a B will be hit. Second question estimate graphically an approximate magnitude and direction for the total force exerted on charge be graphic Coalition We just drove the first force and we draw the second force at the tip of the 1st 1 And then the some of these two forces will be starting at the initial point and ending at the final point. That's all it falls. So a rough estimation. But this is 0.9 U turns. Is this 0.2 to 5 new tons said that would be round wondered a bit more and with the angle something like 30 degrees here. So 60 degrees, they or 120 degrees Let's check. Nice donation. I don't even know the results yet. I'm improvising on the spot, so let's see. Question three. Calculate the total force exerted on charge be magnitude and direction. So this time we are going to add electors at brightly. You think our three step system the first step drew the axis conveniently, for example, but will be convenient. Would be to Joe A. Why access positively downwards. And the next axis positively was left like this. We don't have positive signs. Second step, find the components of the net falls our next forces this one blowing lead. So the X component of the net force we may pull to Has this one got contribution? Net falls on the X axis. If you project it directly that point. So no contribution off the force off A on being in the X axis. What about this one? Well, this one is parallel to the X axis. So yes, fully. I see the contribution is FCB the 900 of respect on it is positive because it's in the positive direction as we have chosen. So this is 0.25 new terms, as expressed in the text as do the Y axis. Now on the Y axis, you see that this one has no contribution because his public why this one has a full contribution. So the contribution of this one on the Y axis is a baby. It's it's magnitude. It's yes, 0.9 u turns. We have now the components of the Net sports so we can carry out Step three, which is to find the magnitude and direction by converting Cartesian coordinates to pull according so we have f equals square it off. The sum of opponents squared. That is 0.9 squared. Plans 0.2 to 5 squared, giving me 0.93. So I was a top team mystic on a graphical estimation of the magnitude. But that was no fun. The angle he just use of tension minus one Formula 10 to minus one of the Y component, which was 0.9 over the X component, which is 0.2 to 5 so that basically for times in minus one off full, I get an angle or 76 degrees in reference to what? To the X axis? Because, yes, component is on the denominator. The X axis is elected that way. The number positive direction for angle will be anti clockwise. So when I go like this, Young was positive. 76 degrees will be that, Yeah, I was also quite optimised by saying 60 degrees now 76 degrees in reference to the positive direction off the X axis. This spaceship is in trouble, its engines of them and it is navigating food. And that's to each field. Three major asteroids exerting gravitational forces on the ship. The first force is 1000 new term, the 2nd 500 in the 3rd 1 1500 The first and the second force are perpendicular to each other, and the 3rd 1 makes an angle of 60 degrees with the 2nd 1 The navigation crew needs your help. They need to know the direction and magnitude of the net force of the ship in order to evaluate if the ship will hit one off the asteroids, they need your help. Were you able to save the crew of the ship? The net force is what you're looking for. So you will have to some these vectors? Yes. And their forces air some off forces which are applied on their body. The first step would be to do a free body diagram, which I did here and at the same time to draw the access. I'm going to use the fact that I know that these two forces perpendicular. I also know the angle of that. Using this fact, I can choose my access along F one F two. So let's define this access. I mean X on this access like being why what we just did corresponds to the first step off a Presidio to add vectors together. Now we are going to do the step to which is to find all the components off these doctors on the X axis. Add them together, and that will give us a component of the net force on the X axis. And if you do probation on the Y Axis XX is F Net Evans, the component of the one on the X axis. Well, it's f one the magnitude of the fund, because if one is ah, no to the X axis and in the positive direction here. Plus what about F two? F two is perpendicular to the X axis has no contribution on the X axis. Plus, what about F three? Well, every makes an angle with the X axis, so we know the angle between F 20 freeze 60 degrees. So here it would be 30 degrees. So the I was in between the director and the access in which you want to project the victor . So it's going to be f me co signed for 30 because you squashed the angle when you projected . But you see that the projection here is in the negative damage. Therefore, you put a minus in front. That's plug in the numbers. If one was 1000 Newtons 00 minus 1500 and co sign off 30 is 0.866 That gives me 300 museums minus so we could actually do it. The X component of the net falls the X axis negative minus 300. So something probably like this. So that would be f X. There's a paedo probation on the Y axes. Definite. Why does everyone have a contribution on the Y axis? No, it's purple eclipses, even for about F. Two have to is aligned with the Y axis, and it's in the positive direction. So local division is its magnitude F three. When you project F three on the Y axis, you actually squash us 60 degrees angle. So giving you f me cost 60 on it is in a positive direction. So you keep the place here. It's plug in the numbers do you owe? Plus, if he was 500 plus F me was 1500 cause I know 60 is 1/2 so that's 7 50 plus 500. So that's 1250 new types so we can draw it from the white access. That's what's 500 said would give you something like this. Well, f y to buy something. Them graphically, we could have an idea already off the net falls affects less if why stop from initial points and point? Hey, that sound that force that's a force on the station, especially, would be going that way. We have the components of the Net force. These are the Cartesian coordinates that three We can find the magnitude in the direction by converting these coordinates to polar coordinates. The magnitude off the net force would be the some all the squares component square with it . Myers 300 squared plus 1000 turned 50 squared. I found 1000 turning in 85 which I'm going to round up to 1300. What about the angle? The direction I use a Formula 100 minus one on the Y component divided by the excrement to that country. Minus one off 1250 divided by miles 300. You see a guy negative sign here. You don't see on camera. I've got a negative sign here, so I'll have a negative angle compared to the X axis. Let's calculated. I found minus 76.5 degrees. That's why we adopted minus 77. So theta equals minus 77 debates. Let's look at our diagram. Take the X axis and we find money. 77 degrees with Lady Village, the X axis. So underneath the X axis is minus start. That would be this. This time doesn't fit. But yes, it does. Because remember the properties of tangent. Here you're Sylvian equation with tangent. And you know that tension Piter is equal to tangent plus pi or plus 1 80 So there's another solution to this equation, and it's this angle plus 1 80 so minus 77 plus 1 18 give me 103. I have an angle 103 that's 103 reads. You could say also that the direction off the net force is plus 13 degrees related to the Y axis. Congratulations. If you followed carefully or lessons off this section and working all the exercises, you're now vector expert. So next time you encounter a physics problem and exercise one exam with doctors, remember what you learned in this course, and you will be fine anyway. Well done.