In this tutorial we are going to talk about vectors the introduction to vectors. Okay, so vectors is Just basically a physical quantity which has got both magnitude and direction So we went first to compare two things. We have a vector and a scalar So we have what we call the vector quantity and the scalar quantity. So what is a vector quantity and what is the scalar quantity? Okay, so a vector quantity like we have already said a vector quantity is a physical quantity which has got both magnitude and direction that's all so that's basically a vector okay what of a scalar scalar is just basically a physical quantity which has got only magnitude now when we are talking about the magnitude we are talking about the size okay so a magnitude is just basically a size now when we're talking about the vectors we are saying that a vector is a physical quantity which has got both magnitude and direction meaning that a vector is supposed to have a size and direction so let's say a car is moving at 20 kilometers per hour south then we say a car is moving at 20 kilometers per hour just like this so the moment when we say a car is traveling at 20 kilometers per hour south meaning we are giving the direction to say this car is moving at a speed of 20 kilometers per hour the direction is toward the south so south is the direction 20 kilometers per hour it is the magnitude okay so if we can see here here we are saying that a car is moving is traveling at 20 kilometers per hour there is no direction definitely there is just only magnitude only size so that is a scalar okay so that is what we need to know the definition of scalar quantity and the um the vector now we have got some examples of vectors so some examples of vector quantity we may talk of um the force Force is a vector quantity.
We may talk of the momentum. Momentum. We may talk of the acceleration.
We may talk of the velocity. We may talk of, we may talk of, what else? We have the force, momentum, acceleration, the velocity and what else? Okay, the load. Okay, we may also talk of the displacement.
Okay, then what are some of the examples of scalar quantities? We may talk of the distance. We may talk of the mass. We may talk of the time.
we may talk of the speed yeah they're a lot okay so these are some of the examples um of vectors and scalars so the difference between scalars and vectors is that vectors is a physical quantity which has got both magnitude and direction while scalars they their physical quantities which they wish only have got what the magnitude which is the size so that is the basic idea behind the scalars and vectors but our main focus in this lesson we are going to talk about vectors only okay so as you can see i've just written the heading there to say the introduction of vectors and scalars so there we go okay so vectors since we are saying that a vector It's a physical quantity which has got both magnitude and direction. Meaning that a vector can be added. So let's say we have a vector. Let's say we have a vector A which is 20 newtons. Let's also have vector B which is 30 newtons.
Let's find a plus b. Now to find a plus b is just a matter of us adding these two vectors. Okay, so these two vectors, we are going to add them.
We're going to get this and then we add the other one, meaning that I'm just supposed to say this is going to be 20 plus 30, which is going to give us 50 newtons. So this is also what we call the resultant vector. So this is similar resultant. The resultant is the addition of all the vectors which you have.
So the addition of A plus B. So now this type of vector is what we call the parallel vectors. So parallel vectors, whenever you have got a parallel vector, you just add them directly.
So as you can see here, we have parallel vectors, you can add them directly. Now, since we are saying that a vector is a physical quantity that has got both magnitude and direction, and a vector can be. can have negative a vector can can have negative but the scalar cannot have negative so we are saying that speed is this is an example of a scalar then velocity is an example of um this is an example of vectors okay so velocity can be negative but speed can't be negative so whenever you see speed having negative that is velocity okay so now we are still talking about the parallel vectors so let's say we have vector a vector a is 30 newtons we have vector b vector b is moving in opposite direction now it is a 20 newtons so to find a plus b this is going to be 30. we follow the arrows because a vector has got both magnitude and direction so the election is very very important where is it going is it toward north toward west towards south toward the east something like that so this vector is going toward the east therefore is going towards west sorry therefore that vector is supposed to be negative okay meaning that this vector is going to be negative we can put plus and then you say it's going to be negative 20. Therefore, the resultant is going to be 10 newtons. That is the parallel vectors. Let's have different things.
Let's say this vector B is 50, 50 newtons. So to find the resultant, the resultant is the addition of A plus B. This is going to give us 30 plus. This is going toward east, toward west, since we know that if we've got our xy plane, this is negative x.
This is positive x. this is a negative y this is positive y okay so this is going toward the positive x direction so we have to take it negative so it's going to be negative 50 and then we're going to get negative 20 newtons what it means there is that when you add these two forces the net force is going to be moving in this direction what it means there guys is this it's more like we have an object let's say we have we have two two guys so we have the surface we have the object now this object has got mass you apply the force in this direction this force is 20 newtons you apply another force in this direction this force is 30 newtons okay so we know that the force that is moving the 30 is greater than me it's greater than 20. therefore this object is going to accelerate in in this direction toward west meaning that the net force is negative what because we're supposed to say that the summation of all the forces uh is going to be the 20 minus minus 30 which is going to give us negative 10 so the net acceleration is going to be in this direction then the net force is a negative uh 30 negative 20 sorry negative 10 okay so that is what we need to understand now let's talk about perpendicular vectors we don't talk of perpendicular vectors these two vectors they are perpendicular to each other so we have vector a which is um 20 newtons let's have vector b so we have our vector b which is moving in this direction it is just basically um let's let's give it a 10 newtons Now to find A plus B, which is going to be equal to the resultant, we need to add these two forces. So we are going to have vector A. plus vector b so our vector a is here b is there so to add these two we need to get the line from the origin all the way to the last vector meaning that this is our a plus b which is the resultant so since we are making a 90 that's why we are saying it is a perpendicular perpendicular vectors how do we add perpendicular vectors to add perpendicular vectors what we have to do is you get the vector for example this case is going to be a plus b or you can just say the resultant it's going we are going to use by the gas theorem where it's going to be the square root of a squared plus b squared okay supposed to say a is a squared plus b squared a squared plus b squared so our resultant is going to be our a is 20 plus our b is 10. so now what would be our answer okay so you can find the answer there 20 squared is supposed to be somewhere there 400 and then we have we have also what what else do we have we have um 10 squared 10 squared is just basically 100 so we can say 20 squared plus 10 squared then we get i'm getting 500 we get the square root of this a 500 square root over 500 so i'm getting a 22.36 okay so my resultant 22.36 newtons So that is it for perpendicular vectors.
So whenever you see perpendicular vectors, but in most cases, you're going to see a question which is going to be neither perpendicular nor parallel. Now here is the issue. We go to the next step. The next step is the vectors which are going to talk about neither perpendicular nor parallel.
Now what you have to understand is that when you talk of neither perpendicular nor parallel, that vector will be. or not be in x if this is my xy plane that vector is not going to be in in x direction or in y direction okay so this vector let's say that we have a vector which is um here is our vector let's say this is a okay by the way i didn't mention to say whenever we're talking about the vector a vector is represented by either a modulus or an arrow on top. So if I have got vector A, I can say A, I put arrow there, or I can put A, then I put the modulus. Okay, that is a vector. When you just leave it just like A, then it's not going to be a vector, it's going to be, it's going to be the scalar and not a vector.
So here is our vector. Let's say we have vector A. This vector A is neither parallel nor perpendicular.
as you can see there is just one vector now this vector is going to have the x component and the y component meaning that this vector is going to have a x and a y then it's going to form an angle here so using soccer tour using soccer tour we can see that we can start with sign if we get the sign there we are going to see that sine theta is going to be equal to the opposite is ay divided by the hypotenuse a so whenever i'm trying to resolve any vector into y component i'm supposed to get that vector i get sine the angle that is the first formula which you have to know and very very important next let's go to cos we have cos theta has to be equal to cos theta is the adjacent over hypotenuse so adjacent is ax hypotenuse is a to find ax is going to be equal to vector a cos theta this is the formula which we're going to be using to find the x component always now this is very very important now let's talk about uh the tan tan is going to help us to find the angle so if you have the ax and ay then we can find the angle So the angle is going to be tan theta is going to be equal to tan theta is the opposite divided by the Adjacent so opposite is a y so we are going to have our a y everything divided by The adjacent which is in a x to find theta is going to be Theta is going to be equal to tan inverse a y divided by a x This is the formula for finding the The direction now in vectors whenever we're talking about the direction we're talking about the theta okay the angle so sometimes you might be given the direction to say south then it's west yeah something like that but sometimes if if they give you the angle the angle is the direction okay very very important now let's say we have um we have a vector let's say we have a vector let's just get rid of this And let's assume to say we have a vector. Let's have vector A, which is going to have the magnitude of 20 meters at an angle of 30 degrees. Now find the X component and the Y component. Now to find the X component, we know that we have made the formula already. The X component is always going to be the A cos theta.
So I'm going to say that my A is 20 because the theta is 30. So we get our calculator. What would be the answer? So we have 20 equals 30. I'm getting 17.32. 17.32 meters.
Let's get to find the a y, the a y is going to be a sine theta. So this is going to give us our a is 20 sine 30. This is going to give us 10 meters. Okay, so this is the basic idea behind the introduction to vectors.
Okay, so in this video we want just to know how we can find the direction of a vector. Okay, so in the previous video we talked about how to find the angle and everything. Okay, and for just a lookup we said that if we have got a vector which is lying in the first quadrant, all the vectors are supposed, all the angles are supposed to come from positive x-axis. Meaning, if I have 20 here, I'm supposed to get the angle from positive x axis.
So it's supposed to come from here all the way to this line, meaning that for this vector, I'm going to use 20 to resolve it. So all the angles are coming from positive x axis. That is what we said.
So now let's talk about how to find the same direction. Okay, that is also very, very important. We need to know.
So let's have our xy plane. Okay. And then remember, we are going to have This is going to be our north. Okay, so we have our north, our east, our south, our west. So we know that this is the first quadrant.
This is the second quadrant. This is the third quadrant. And this is our fourth quadrant.
Now we know that this line here is positive x-axis. This line is negative x-axis. this line is positive x-axis this line is negative x-axis that is very very important guys now whenever you are resolving vectors into x and y component remember you are going to be adding those components now you discover to say you have got the x component which is positive and the y component is also positive Now, where do you expect this angle to fall? Definitely, this angle is going to fall in first quadrant. Why?
Because in the first quadrant, that's where we have the X is positive, the Y is positive. So this line, that angle is going to be in the first quadrant. Now, after using the formula, remember our formula was theta was equal to tan inverse.
the let's say we have is y the y component divided by the x component after plugging in the values using this formula let me let me write it here the x we have tan inverse we have the y divided by the x so after plugging in the values using this formula this formula what we have to understand is that in that angle is going to fall in the first quadrant now after finding the theta that theta is going to be the answer so theta is the answer okay now let's say that the x component you discover to say the x component is a negative the y is positive let's check so we can see that the x is negative is this line the y is negative is positive is this line meaning the angle is supposed to be in second quadrant now in second quadrant after plugging in the values now what i have to remind you is that whenever you are using these principles you ignore the negative don't plug in x as a negative just ignore the negative and then after ignoring the negative follow these principles which i'm going to tell you so you're going to say that 180 minus the theta which you're going to find after plugging in the values here okay So you're going to see some examples which we're going to have. So now let's go to the next one. We have, let's say we have the x is negative. The y is also negative.
Definitely the angle has to fall in the third quadrant. Third quadrant, we have got the x negative, the y positive. The angle is supposed to be in this quadrant. So in the third quadrant, what we do is 180 plus the theta. Remember you ignore the negative.
plug in as a negative here just ignore any negative which you see okay so the fourth one is going to be when the x is a positive the y is negative so the x is a positive the y is negative definitely has to fall this angle has to fall in the fourth quadrant in fourth quadrant we have this positive this negative we have negative the y you the x positive so it's going for here so now what we need to understand that when it falls in the third fourth quadrant is going to be 360 minus minus theta the one which you have found after plugging in the values there that is very very important now let's get to see some examples we see how we can solve some certain examples let's say we have the angle let's say we have the the vector let's have two vectors okay let's have two vectors we have vector a which is 20 20 meters at an angle of 40 degrees we have vector b which is lying which is 40 meters at an angle let's say it lies in um toward the positive x-axis okay now to resolve these two vectors it's very important for you to come up with um you come up with a an xy plane so xy plane is going to be here so we're going to have our vectors we know that the first one is 40 degrees so 40 degrees we know that these vectors they're always measured from positive x-axis is going to be here okay it's going to be it's going to be here so here is going to be our 40 it's going to be our vector a which is uh 20 meters so the angle is going to be 40 degrees another one is uh we have vector b which is lying toward positive x-axis meaning the angle is zero so to find the question is how can you find the the resultant So to find the resultant, since we're talking about the vectors, as long as they're asking you to find the resultant, they're asking you to find the magnitude and the direction. So to find the magnitude, first we have to resolve each vector. This is vector B, which is 40 meters.
So we have to resolve each vector into X and Y components. So we're going to say that the AX is going to be, we know that resolving a vector into X is going to be A cos theta. So AX is going to be equal to, our A is 20, the vector, because the angle is 40. So our AX is going to be, we are going to have 20 equals 40. which is going to be 15.32 15.32 meters so that is the x component let's go to y component ay is going to be a sine theta to resolve any vector into y component is sine so it's going to be um ay is going to be the a is 20 sine 40. so our ay is going to be equal to we have 20 sine for 12.886 i'm getting a 12.86 i've just landed off is 12.8551 one or seven five so i've just landed off so to create space i can put these here i have a y i have a x as um 15.32 a y as 12.86 cool now let me get rid of this let's go to vector b vector b we have um we're going to say that b x is going to be equal to vector b cos theta so the b is a 14 because the angle is zero degrees so what is 40 equals to zero It's going to be 40 cross 0. I'm getting 40. So my BX is 40 meters. Then I go to the BY.
It's going to be B sine theta. It's going to be B sine theta. So it's going to be BY.
It's going to be is a 40 sine 40 oh sine 0 So this is going to give us 0 It's going to give us 0 Now we have the x so now what we need to do is the question is we want to find the resultant Let's put them here the a the by the bx is uh 40 the by is zero let's get rid of this now after resolving each vector into x and y component we need to add the x components alone and the y component alone for us to find the resultant so it's going to be the resultant for x component is going to be the ax plus the bx so the rx is going to be this is the the x component of the resultant which we are trying to find so it's going to be the ax is basically 15 0.32 plus the bx is 14 so we are going to have rx is going to be equal to 40 plus 15.2 0.32 so i'm getting 55.32 meters that is the x component of the resultant let's go now to y So we're going to have the y component is going to be ay plus by. So let's plug in the values. The ay is a 14. Oh, the ay is 12.86.
The by is 0. So our ay is going to be 12.86. Okay, so that is our one. So let's just do this.
We can just get rid of this now. and then we want to find okay let's just get rid of this we want to find the resultant remember so to find the resultant let's just put the rx we know that rx is 55.32 meters want to find again we have found the ry as 12.86 now we want to find the resultant after getting the after you add the only issue here is as long as you have been given maybe more than one vectors you have to resolve each vector into x and y component then add the x components alone add the y components alone okay so now after adding them meaning you have now what you have now is this this is the resultant we're trying to find you have now the x component which is the rx and the y component which is the ry Meaning we have gone back to the Pythagoras theorem. So here we are going to say the resultant is going to be equal to the square root of the rx squared plus the ry squared so this is going to be so the rx is 55.32 squared plus the ry is 12.86 squared okay so we get our calculator and then we find the value now that will be our magnitude so it's going to be 55 0.32 i squared plus 12.86 i squared i'm getting um now i need to square root it the answer has to square root it so i'm getting 56.79 which is the same as just 56.8 so my resultant is going to be 56.8 Let's just put 56.79 meters. So that is my resultant. But the moment when you leave the answer just like this, meaning this is not a vector.
A vector can't stand alone without the magnitude. Sorry, without the direction. So this is just the magnitude. We need to find the direction. Area 1, we say the vector quantity is a physical quantity which has got both magnitude and direction.
So we need to find the direction. Now, to find the direction, remember, we say that to find the direction we use the formula which is going to be the theta is going to be equal to the tan inverse so it's going to be the ry divided by rx then it's going to be the theta is going to be equal to tan inverse remember we don't have any negative but if we had negative maybe it was negative would have ignored the negative and follow the principles so ry is at 12. point eight six the rx is 55.32 let's plug in and find the value of x so it's a shift you press where there's shift and then it's going to be 10 it's going to be shift you press where there's 10 then you open the brackets it's going to be 12.86 divided by 55.32 cross the brackets So I'm getting 13.08, which is just the same as 13.1. So 13.1 degrees. That is what I'm getting.
Now, after finding the theta, you are not done. After finding the theta, we'll go back to the principles now where we said if the X is positive, remember here. Now I'm talking about the resultant.
After finding here, we can see that our Rx. was positive our ry was also positive here is positive positive so we go back we say that if ry is positive the x is also positive definitely the angle has to fall because this is z positive y this is positive x the angle has to fall in this quadrant the first quadrant so the first quadrant the theta which you have found that happened to be your answer meaning that this is our final answer okay very important So if it was negative, let's say that the rx was negative. So if the rx was negative, meaning that the angle is supposed to fall in this quadrant, the rx is negative, meaning the y is positive.
It's supposed to fall in this. It's supposed to fall in this. And then you say 180 minus the theta.
That would be the final answer. Okay, so you get to see how we are going to be solving the equations because next chapter we are going to talk about how to solve the questions but before we do that we need first to know how to sketch the vectors okay thank you for watching this video see you in the next lesson where we're going to talk about how to sketch the vectors okay okay so in this video we are going to talk about sketching of vectors okay so i hope you managed to watch part three where we talked about um how to to know that the angle is supposed to be in the first quadrant second quadrant and the fourth quadrant okay we also talked about um how to dissolve vectors into x and y components something like that now here we want to talk about discussion okay so sketching if you have been given the vectors let's say you have three vectors let's say you have vector a which is uh 20 meters at an angle of 30 degrees then you have vector b which is 30 meters at an angle of let's say 210 degrees then you have vector c which is um let's just say 35 meters at an angle of um at an angle of um maybe three or let's just say at an angle of um two let's say 115. degrees okay cool so now how do we sketch these vectors so it's very very simple the first thing to do is you need to come up with x y plane okay so i'm going to come up with my x y plane here so here is going to be my x y plane it's either you can have a visible one or you can have a dated one so i'm going to have a visible one i'm going to have just a small one so i know that this is minor this is my south this is my east this is my west so the first vector i'm going to assume that this is my origin that is my starting point so the first vector is supposed to start from there all the way to 30 degrees so another 30 degrees sum is near east than north so i'm going to put it here say that is going to be is 20 so i can put just like this so that is going to be now at the end there i'm going to put an alarm so i'm going to say that that is my vector a which is at 20 meters now from here all the way to there i know that i can even put the angle there that is 30 degrees now after this is our end point of vector a now from this end point i'm supposed to write another dotted form so i'm going to put another x y plane so i can even put a dotted one it's okay another x y plane so i know that this is my positive x-axis we know that vectors when we're measuring the angles they are all coming from positive x-axis okay now this is going to be my vector b so vector b is 2 10. so where is 2 10 is 2 10 in first quadrant is 2 10 in second quadrant is 2 10 in third quadrant or fourth quadrant okay 2 10 is in state quadrant so we know that this is our first quadrant second third fourth meaning that this 2 10 is in this quadrant so i'm going to start from there I'm going to say that from this line, I'm going to write, I'm going to have another line from there. Then it has to go in this direction. Okay. So that is 30 meters.
So 30 meters is supposed to be a bit longer than the vector A. So B is going to be a bit longer than vector A. Okay. Then I'm going to say that this is going to be my end point of vector B. So I'm going to say this is my B and it is 30 meters.
Now what I'm going to do is after reaching at the end point I need to come up with another x y prime Okay, so I know that this is going to be positive x this is going to be positive y this is negative x This is negative y again. This is going to be my starting point of vector vector c So vector c is going to start from there instead 5 meters. So it's going to be bit longer than than vector Vector a and b so it's going to start from here from this line now this 115 115 is in the second quadrant so it's going to be in this quadrant but is it going to be near is it near uh is 115 near north this is our north okay this is our north and this is our our south this is our west so is it near west or north okay so how do you know that it's near so this is 150 we know that this part here is 90 meaning 115 is supposed to be we know that from this line all the way to this line is 180 meaning that 115 is near north so I'm going to say that this is going to be it's going to start from there all the way until so it's supposed to be a bit longer than all the vectors okay so I can put it here so that is my end point of vector c which is 35 meters Now to find the resultant, resultant is supposed to be the connection, the addition over from the first point of the first vector.
This is the first point of the first vector until the last point of the last vector. So that is going to be our resultant. So you can say that that is going to be our R. Now it's either you can resolve this vector and find the magnitude and direction. now from this line here you can see that this resultant it is lying in second quadrant this is the second quadrant so even if you you resolve this vector into x and y component you are going to discover that the magnitude of this vector is going to fall in second quadrant okay so that is caching now let's have another example let's have let's have four different vectors let me just include another one let's say we have another vector d which is um 40 let's just put 25 now let's put let's change the angles the directions so let's have this one is due let's say this is due due east then this is at an angle of 30 degrees let's say 35 degrees this is at an angle of we put due south or due west okay it's due west then this is at an angle of 3 30 degrees let's now sketch these vectors so the first thing to do is we have to put we have to put the xy plane here so i'm going to have my xy plane here just a small one if you want so this is going to be my origin i know that this is north this is east this is uh west south this is west so it's supposed to start from the origin all the way to this point here now we're supposed to to move 20 meters so i'm going to say this is going to be my vector my vector a okay so it has ended there so that is my a which is uh 20 meters we know that when you say do east the angle is zero so it is in in east side okay now this is going to be my first point now for another vector so i'm going to put an x y plane again here so this vector they're saying it is going 35 meters oh 35 it's 35 meters at an angle of 35 degrees so it has to start from this this is going to be our origin now 35 is it near now north or east it's near it's near east so it's going to be this is supposed to be in this direction so it's supposed to be a bit longer than vector a okay so it's going to be somewhere there so say this is 30 so we end there that is going to be our vector b which is 30 meters so you can even indicate the angles if you want you can say that from this point you say this point here you say this angle here is what is 35 degrees then we have reached now here you make another xy plane so we have been told that this vector is due east due west West, this is north, this is east, this is south, this is west.
Meaning that this angle, this vector is lying in this direction. So it's that 5. So 5 is supposed to be a bit longer than B, so it's going to go there. Okay, we make it a bit longer.
So we end there. Again, after reaching there, I make another xy plane. Because I have another vector.
so i know that here i've made this one this is also east this is north this is south this is uh this is west this is three feet so where is three feet three thirty is uh in this line it's supposed to be in this line but it's going to be near it's supposed to be near east than than uh south okay so i'm going to start from here so i'm going to start from there then i say this is 25 although 25 25 is going to be just a bit longer than 20 okay so i can even finish it there so i say this is my my uh b my c oh it's my d sorry which is a 25 okay then this angle from here all the way to this side is going to be three 30 so to find the resultant resultant is the connection from the first point until the last what from this point first point to the last vector so this is going to be my resultant so if you happen to find this vector you resolve it into x and y component and find the resultant you are going to see that the answer which you are going to find the angle is going to fall in the fourth quadrant as you can see this is the one we are looking for so it's going to be in the fourth quadrant Okay, so this is how you do or this is how you can sketch vectors. It's not complicated. So you can just change some values there and then you try you try it out on how you can sketch it. Okay, so now from here, let's try to Let's try to resolve this vector into x and y component and then let's find the resultant.
From here, how do we find the resultant? So to find the resultant, we are saying that this angle is supposed to be in fourth quadrant. So let's see if it's going to fall in the fourth quadrant.
Fourth quadrant. Okay, so let's resolve it into x and y component. Let's start with vector a. So we know that we said resolving a vector into x and y component, we have to...
we get the vector we get the vector okay we get the vector then the for example this is a so it's going to be a x is going to be equal to a cos theta so this is going to be a x is going to be equal to a is 20. then due east the angle is zero so it's going to be cos 0. so a x is going to be equal to because uh 20 cos 0 we are going to get um we're going to get a 20 okay so that is the x component let's let's have our y component so y component is going to be a sine theta so ay is going to be equal to our a is 20 sine 0 so sine 0 is going to be 0 0 times 20 we are going to get a 0 so i can just put it here to say we have our ax okay we have our x as 20 ay 0 let's go to the next vector we have vector b this is all vector b into x and y component we want to find the resultant so it's going to be uh b x is going to be equal to b sine b cos theta so it's going to be 30 cos 35 Okay, so let's see what we're going to get so 30 we have 30 then of course 30 plus 35 says 25 point five seven Okay, so I'm going to put it here BX is 24 point five seven. So let's see our BX oh our by sorry so by is going to be b we know that in y in y is going to be b which is 30 sign 35 so we have 30 sign sign that 5. i'm getting 17.21 okay let's go to The next vector let's go to vector vector C so Cx is going to be equal to that 35 Because due West is 180 the angle is 180. So this is going to give us a negative 35 So our B our Cx is going to be equal to negative 35 then our CY is going to be equal to Our CY is going to be CY is going to be C is 35 and then sine 1 8 is going to give us 0. Okay, so from there then we go to D where we'll be able to talk about DX DX is going to be 25 plus 3 30. So DX is going to be 25. because um because 330 so i'm getting to 21.65 21.65 let's talk about dy so dy is going to be a 25 sign d y is going to be 25 sign 3 30 so 25 sign 3 30 i'm getting negative 12.5 okay now after getting these the x component we now get rid of this we no longer need this we want to find the resultant so we need first to find the x we need to add the x components alone the y components alone okay so we're going to say that the x component or the resultant is going to be It's going to be AX plus BX plus CX plus DX. So our RX is going to be equal to our AX is 20. Our BX is 24.57 plus our CX is negative 35 plus our DX is 21 21.65 so we can add these values and then we are going to see that we are going to have 20 plus 20 plus 24.57 five seven 24.57 plus it's going to be minus 35 plus 21.65 okay so i'm getting 31.22 31.22 is meters so we can get rid of this we can just put our resultant there on top let's just put it here our rx is 31.2 meters now let's find the y component of the resultant so it's going to be it's going to be the a y plus b y plus c y plus um d y So we plug in the values We can see that AY is 0 BY is 17.21 plus CY is 0 plus negative 12.5 So our RY is going to be what? Our RY is going to be it's going to do what it's going to be 17.21 and then we have 17.21 plus open brackets negative 12 I'm getting 4.71 meters So to find the resultant To find the resultant is going to be the resultant is going to be equal to the square root of rx squared plus ry squared so we can now plug in the values let's just put our resultant here the y component which is 4.71 okay so we are going to have this we're going to have um we plug in the values okay so it's going to be uh 31.22 squared plus 4.71 squared so our resultant is going to be 31 0.22 squared plus 4.71 squared i get the square root i'm getting that 1.57 at 1.57 meters that is the resultant now to find the angle we know what to do so i guess we have made a mistake somewhere because the other y was supposed to be negative so that the answer the answer was supposed to be in fourth quadrant so anyways okay so let's just go ahead and find the angle so it's going to be theta it's going to be equal to tan inverse ry rx so you plug in the values and find the value of what the value of theta okay so that is it for sketching i hope you have enjoyed okay so let's continue now we are talking about vectors Okay, in the previous video we came up with the formulas on how to come up with the x component And we say that if for us to find the x component if we maybe we have got a vector a So we say that if this vector a is supposed to be if we want to find the x component It's going to be vector a cos theta then the y component is going to be the a sine theta Then we say that if we want to find the the the direction which is the theta the angle so it's tan theta is going to be equal to a y divided by a x then theta is going to be equal to tan inverse a y a x okay so this is the basic idea which we have to understand under vectors okay now let's say we want to we have a vector So first of all, we want first to talk about the angles. How can you choose the angle?
So let's have our xy plane. Okay, here is our xy plane. Okay, so here is our xy plane. Then let's say we have a vector.
Let's have vector a. Okay, that is our vector a. Now, maybe we have been given the angle.
to say this angle here is 20 degrees to resolve vector a into x and y the angle which we are going to use is going to be the 20. so as long as we are talking about the vectors we are measuring the angles from positive x axis so this is negative this is negative y this is positive y so to get the angle which you are supposed to use when when plugging in the values here to say this is going to be Corsi so this filter is going to be we are getting the angle from this line all the way until it touches the vector So this line all the one to touches the vector is 20 meaning that 20 is going to be the angle So I'm going to use okay, and let's say that we have a vector where the angle is here We have the vector there then we have got our angle here 20 degrees So which angle are we going to use for us to resolve this vector? Remember, we are getting the angle from positive x until it touches the vector. Meaning that from this point here all the way to this line, it is 90. So I suppose to say 90 minus 20, which is going to be 70. Meaning that the angle which is going to be here is going to be 70 degrees. Okay.
Now let's have different things. Let's have a different thing. Let's say we have the angle. a vector which is somewhere there okay then we have been given that here is 40 degrees so to resolve this vector into x and y component we are supposed to get the angle the angle which we're going to is supposed to come from this line until it touches that line okay so we know that from this line all the way to this line it is 180 okay so we're going to say that 180 180 minus 40 Okay, so the answer is going to be 140. So the angle which we're going to use is going to be 140. Now, let's say we have the angle here. Let's say the angle is here.
We have vector A. Then here is the angle. The angle there is 20 degrees. So we know that for us to resolve this vector, it's supposed to come from this line until it touches this line.
So we know that from this line all the way to this line, it is 90. So 90 plus 20. supposed to be 110 okay then let's say we have another vector which is lying in third quadrant so let's say the vector is here this is vector b want to resolve vector b into x and y component and let's say that the angle which is here is um is uh 30 degrees remember we want to get the angle from this line okay from this line all the way until it touches this line so we know that from this line here all the way to this line is 1 8 so 180 plus 30 is going to give us what um is going to give us 210 okay so that one you have to know now let's say we have another another thing which is here let's say the angle is here here is the third thing so we have the third degrees there and then we want to get the angle which is going to help us to resolve this vector into x and y component getting the angle from this point all the way until it touches this line we know that from this point all the way to this line is 180 then we know that from this point all the way to this point is 90. so we're going to say 90 minus 30 which is going to be 60 meaning this part is 60. so it's going to be 118 plus 60 degrees so it's going to be 180 plus 60 is going to be 240. so the angle which we are going to use is going to be 240 degrees let's now go to fourth quadrant we see what we are going to be having okay so let's say we have let's say we have the angle which is going to be here okay then we want to get the angle let's say this angle is 30 20 degrees let's just say 20. so we know that it resolve this vector into x and y component the angle is supposed to come from positive x-axis so we are measuring the angle and clockwise they are going in this direction and not clockwise the moment when you go clockwise meaning you are talking about negative angles now so we are starting from this line until all the way it touches this line so we know that this point here is 270 we know that this is 90 this is 0 this is 180 this is 270 then again when we reach at this point from here all the way to this line again it's going to be 360 okay so if a vector is um is in fourth quadrant and then you have been given the angle this is going to be 270 plus 20 which is going to be 290. now if you have been given the angle here let's say the angle is uh is here 20 let's say 30 so we're getting the line from here all the way until it touches there so it's supposed to be we know that the full circle is 360 is going to be 360 minus 30 which is going to be 330 so that's how we get the angles actually very very important you have to know that okay now from here we need to let's also talk about Something else the same thing but a different one. We have the xy plane here is our xy plane Now as we we are saying that we are making the angles from positive x axis. This is positive. This is negative x this is um negative y, okay This is positive y So what we have to understand is that When you have been told that a vector is lying toward the positive x-axis or this is the same as east this is west this is south oh this is north this is south if a vector is going toward the positive x-axis meaning it is going toward east sometimes they might tell you to say a vector is going toward east or a vector is moving in positive x-axis okay Whenever you see that it's positive x-axis, meaning that vector is in this line, that vector is in this line, and that vector, the angle is 0. So the angle is 0 when the vector lies in positive x direction.
Then when a vector is going towards positive y direction or north, the angle is 90. so here is 0 here is 19 here is 0 here is 90 here is 180 here is a 270 so if a vector is going toward positive negative x axis or west the angle is a 180 if the vector is going toward south or toward negative y direction the angle is 270 that one you have to know okay very very important now from here let's talk about um let's talk about the terms which they use in vectors so in vectors you are going to find that you're going to come across these terms so we are saying that this is north this is south this is east this is west now the terms the first terms which are going to come across is um uh North maybe you have got vector a which is going north or east north or west south of west south of east south of west sometimes it might be east or south yeah it's the same thing east or south so now whenever we are talking about this let's say we have got the angle we have got vector a which is um vector a is um 20 meters at an angle of 40 degrees now they're saying that it is north of east what it means is that this angle is near east than north So if we check here we have got north and east. Meaning this 40 degrees is near north than east. So meaning that here is the angle.
It's going to be like somewhere there. So here is our 40 degrees. This 40 degrees is supposed to be near.
Just a minute. This 40 degrees is supposed to be near east than north. So it is near to the last part than the first part.
So this 40 is going to be here. Meaning when resolving this vector into x and y component, I'm going to use the same angle, which is going to be 40. Because we are measuring the angles from positive x axis. So it's going to come here until it touches the vector.
Okay, now let's say we have the same one. Now it is north of west. Meaning that the angle is near west than north.
Meaning that the angle is here. Is here. This 40 is here. So to resolve this vector, we are going to get the angle from this point all the way to this line, meaning it's supposed to be 180 minus 40, which is going to be 140. Okay.
Then we have south of east. So as long as you see of north of east, south of east, north of what? So the last part, meaning that that angle is near to the last part than the first part.
okay so let's say we have a vector a which is with 20 meters at an angle 40 degrees south of east south of east meaning this is there the angle is near east than south so here is going to be our angle but when resolving this the angle is supposed to come from this line here all the way until touching the vet so it's going to be 360 minus the 40 degrees that is very very important Okay, now let's say we have this. Let's have just a different one, just a bit different one. Let's get to have a different one. Let's say we have this is north, south, east, west.
So we have the angle which is 20 degrees west of north. The angle is near north than west. Meaning this 20 degrees is here.
Here is the 20 degrees. It has to be near north than west. So when resolving this angle, I'm supposed to get the angle from this line until it touches that line.
So meaning from this line to this line is 90. 90 plus 20. so it's going to be um 110 that you have to know okay so the angle is near to the last part than the first part now let's say you have been told that you have vector a you have vector a which is 20 30 degrees clockwise clockwise i'm saying that the angles on vectors we are measuring them and clockwise they are supposed to go in this direction whenever you see clockwise meaning the angle they have measured it using in that direction meaning it is going toward negative meaning that angle is starting from here until somewhere there that would be our that would be our way set so to get the angle which we're going to resolve this it's going to start from this line until it touches this line it's going to be 360 minus minus 30 or you can even put you can just say this is going to have a negative which is going to be the same if you plug in um let's say we have 20 the the the magnitude is 20 20 equals 330 degrees and 20 equals negative if you put negative since it is going in opposite direction negative 30 we are going to get the same answer but i encourage you using this okay so that is very very important now next let's talk about um the second terms which we're going to come across so we have our xy brain okay so here is our xy brain let's let's continue we have the north we have the north we have the south we have the east we have the west now here is the issue guys uh you are going to come across some certain terms they are going to tell you that we have angle the vector a which is 20 meters no they're going they're not going to give you the angle but they're going to tell you that it is line north east sometimes they can say north west and then they can say south east then they can say southwest so as long as there is nothing in between here meaning that that angle is in between so this angle it is 20 degrees this 20 degrees is in between north and east so the angle is going to be this vector is lying here it's going to be between so you know that here is 19. so what uh what number is between 90 meaning this part is 45. this is 45 to resolve this angle in this uh this vector into x and y component we are going to get the angle from this line until it touches that line so it's going to be 45 now let's talk about north we're talking about north of north northwest so northwest meaning the angle is here so it's going to be both 45 45 45 45 but we know that to resolve this angle we're supposed to get the angle from this To solve this vector, we're supposed to get the angle from positive x-axis until it touches the vector. Simple, right? Yeah.
So it touches the vector, then we are going to see that it's going to be 90 plus 45. Just like that. Let's go to the next one. We are talking about now is southeast. So southeast is going to be in between here.
So we have 45, 45. The angle is going to be from this line, it touches this line. 180, we know that from here to there, 180. 180 plus 45. that will be the angle to use okay now next we are talking about um we're talking about uh southwest southwest is going to be uh oh this was southeast which was supposed to be here so this is 45 this is 45 this was southeast i made a mistake i talked about southwest instead of southeast so southeast is what we have there so it's going to be from this line all the way until there so it's going to be 270 plus 45 which is going to be the same even if you say 360 minus 45 it's the same thing okay so southeast southwest i've already explained it's in this one 45 45 you get the angle there so that is the basic idea behind vectors it's very very important for you to know how to get the angle which angle we're going to use to resolve this vector into x and y component very very important next lesson we are going to talk about how to know to say this angle falls in this quadrant this quadrant this quadrant and how to get the angle and how to represent them and also how to sketch the vectors okay thank you for watching this video see you in the next lesson so let's talk about sketching and free body diagram So in this case, this FBD is just basically free body diagram. Okay, so what's the difference between sketching and free body diagram?
So we're going to talk about this in difference to vectors. Okay, so let's have three vectors. Let's have vector A. We say that vector A is 20 meters at an angle of 30 degrees. Let's say we have vector B.
which is 35 meters at an angle of 155 degrees. Let's say we have vector C, which is... 10 meters at an angle of 320 degrees. Now, when you've been asked to say, sketch the vector, or light the free body diagram, those are two different things.
So let's talk about free body diagram first. Now, when it comes for free body diagram, a free body diagram is something that you yourself who is representing that way can understand. But sketching is giving the information exactly the way they are. Okay, so now when it comes to free body diagram, for example, here we have three vectors.
How can we come up with a free body diagram? Now, when it comes for vectors, a free body diagram can be represented by x, y plane. Okay, so we have, let's say we have an x, y plane.
Here is the x, y plane. Okay, so I'll put it here. Here is my xy plane.
So we do know that we have, we're going to have the north, the south, the east and the west. The first vector is saying, which is vector a, is saying 30 degrees. So we know that vectors, they are measured from positive x axis.
Meaning that this first vector which we have here is supposed to start from here all the way to there. So we have 30 degrees. So here is going to be my 30 degrees. Then I'm going to say this is my vector A. So I'm going to say this is my vector A which is 20 meters.
So I'm going to put even the angle there to say my angle is 30 degrees. Now vector B is 155. Where can we find 155? 155 is found in the second quadrant. So I'm going to get my I'll go to my second quadrant and I'll say that this is going to be my 155 Okay, and I'll say that is my vector.
What that is my vector b, which is um 35 meters Now we know that we are measuring the angles from positive x-axis until it touches the vector So it's going to start from here all the way until it touches there So that would be the angle which we are talking about which is 155 degrees Vector C is 320 degrees. So 320 is found in the fourth quadrant. So I'm going to go there in the fourth quadrant and say this is going to be my vector what? My vector C.
Okay. So that is going to be my vector C, which is 10 meters. So we know that the angle is supposed to come from the first quadrant, from the first quadrant, which is the positive x-axis, and all the way until it touches the vector.
So that is going to be our angle which is going to be 320 degrees. So the diagram which I've given you here, this is a free body diagram. Okay, now when it comes for sketching, sketching is a bit different.
So here's going to be the sketching. So we're going to have, you first need to have an XY plane as well. That is going to be our starting point. So I'm going to have my x y prime here.
Okay, and I'm going to say that this is going to be my first point. Okay, so that is going to be my first point. So that first point there we have we have the vector which is vector a which is 20 meters.
So even when it comes for sketching, this is 20 meters, this is that five meters, this is 10 meters, even the The vectors themselves, they are going to differ in terms of the length. Okay. So 20 is going to be a bit longer.
It's going to be longer than C, which is a vector A is going to be a bit longer than vector C. And vector B is going to be a bit longer than A and C. So we're going to start here to say, this is going to be vector A.
So vector A start from there. So it's 30, it's 20 meters. So I'm going to put there to say, that would now my destination.
I'm going to say this is my vector, my vector m, which is 20 meters, I can even put the angles to say from this line here, all the way to that line is 30 degrees. Okay, so I'm going to put that for that is that degrees, that is my destination. So from there, now I'm going to put another dotted line, that dotted line is going to help me to know Where am I supposed to go?
So I'm going to put just a dotted line to put an xy prime. And then I'm going to come there and say, I'm going to vector b. Vector b is 155. So here we have the x.
We have 1, 2, 3, 4. So here is the second quadrant. We know that this vector is found in the second quadrant. Okay, so we're going to go there and say this is going to be our vector, but it's 35 supposed to be longer than the vector a. So as you can see, it's a bit longer. So that is my 35 there.
She's going to be I'm going to say that that is my vector. That is my vector B, which is that five meters. So again, that is my destination.
I've reached now I'm going to make another x y prime just a small Okay, I know that this is 1, 2, 3, 4. This is my fourth quadrant. And I know that vector C is found in the fourth quadrant. So I'm going to go in the fourth quadrant and say this vector.
But it's 10 meters. It's supposed to be a bit shorter than vector A. So it's going to be shorter than vector A. So it's going to be there.
Then I'm going to put it there to say that is my vector. What? That is my vector C, which is 10 meters.
So if I want, I can say. This vector here, from this line here all the way to this line, that is 155 degrees. Then here I'm going to say from this line all the way to this line, that is going to be my 320 degrees.
Simple, right? Then we need now to find the resultant. The resultant is the connection from the first point, from the origin all the way to the last vector.
So here was our origin and then the last vector is this one. So that connection there. that is the resultant vector so that is going to be my resultant vector r now resultant vector is the addition of vector a plus vector b plus vector c okay now when the question is asking you to come up with a free body diagram you need to come up with this so this is a free body diagram and then if they ask you to sketch this is what you're supposed to do that is sketching so that is the difference between and free body diagram.
Okay. In this tutorial, we are going to talk about practice questions under vectors. I've got a number of questions with me here. So let's start with this one. What are the X components and Y component?
of a vector a in xy plane if the direction is 25 to 50 degrees counterclockwise from positive from positive direction of the x-axis and its magnitude is 7.3 meters okay so let's have our xy plane so this is our xy plane we do have the vector which is 7.8 7.3 meters they are saying counterclockwise so counterclockwise is the same as and clockwise okay so meaning that the vector is starting from positive x-axis until it touches the vector so the vector is going to be 250 is in third quadrant so it is here this is our vector here so this vector is 7.3 meters the angle from this point all the way to this point is 250 degrees okay now they want us to find the x component and the y component remember to find the x component of the vector we get the vector cos theta so in this case we have got 7.3 is going to be 7.3 cos theta so this is going to be our x component so x will be equal to seven point 3 cos 250 degrees so our x component is going to be what is 7.3 cos 250 so this is giving me negative 2.49 which is the same as 4 5 okay now it is in meters is going to be in meters as well the y component is going to be 7.3 Sign the theta which is 250 so we're going to have the y component. We have 7.3 sine 250 degrees So I'm getting negative 6.85859 which is the same as 86 meters. So these are the X and Y components of the vector. Now we need to find again the direction.
To find the direction that is going to be theta is equal to tan inverse then you have got the y component divided by the x component so in this case remember if the x component is negative and the y component is negative we expect our angle to be in what to be in third quadrant okay so the question is saying what are the x and y components so in this case we already have the direction which is 250 So there's no need of us finding the direction. So we're only required to find the x and the y components of this vector. So that is it for this question.
Let's go to the next question. So our next question is the displacement vector R in the xy plane is 15 meters long and directed at an angle of 30 degrees as shown in the figure below. Determine the x and y component of the vector. To find the x component of this vector, we're going to say the rx is going to be r cos theta.
So rx will be, what is our r? In this case, our r is 15. The angle is 30. So our rx is going to be 15 cos 30, which is going to be a 12.99, which is the same as just 13.0 meters. So that is the x component of vector r.
To find the y component of vector r, it's going to be ry is going to be equal to r sine theta. So we're going to have ry, the r is 15, then you have sine, the theta is 30 degrees. So our ry is going to be 7.5 meters. So the y component of vector r is 7.5 meters. So these are the x and y component of vector r.
Question 3. The x component of vector A is negative 25 meters and the y component is positive 40 meters. Part A, what is the magnitude of vector A? Part B, what is the angle between the direction of A and the positive direction of x?
Okay, so in short, there are part 1. A, which is part A, is asking us to find the magnitude of this vector, A. Then part B is asking us to find the direction of vector A. So to find the magnitude, we know that A is going to be the square root of AX squared plus AY squared.
So our vector A is going to be... Our AX in this case we have the X component which is a negative 25 we square it plus the Y component is 40 now we square it. Our A the vector A is going to be negative 25 which is going to be 25 squared plus 40 squared. I get the square root I'm getting 47. So in this case, 47.17 meters, this is my magnitude.
To find the direction, first we have to identify in which quadrant we expect this angle to be. Okay. Now we do know that our x component in this case is negative. Our y component is positive. Where do we expect our angle to be?
If I have got my xy plane, I can see that this is positive x. This is negative x. This is positive negative y in that direction.
OK, that direction of negative y, then you have got a positive y. So if x is negative, this is x negative, then y positive is this line. We expect our angle to be in the second quadrant.
If the angle falls in second quadrant using this formula theta is equal to tan inverse. Ay divided by Ax, we are going to plug in positive values in this formula. Then, if the angle falls in the second quadrant, we are going to say 180 minus the theta which we are going to find, provided that we have plugged in positive values only here. If we discover to say the x component is negative, the y component is positive, meaning that using this formula, if both are negative in short x component is negative y component is negative in this formula plug in only positive values ignore the negative then you are going to say if it falls in the third quadrant you're going to say 180 plus the feeder which you're going to find okay then if the angle falls in this first quadrant meaning that the x component is positive the y component is also at positive you get the same filter the features you're going to find using this formula that is your answer if the angle falls in the fourth quadrant meaning that The x component is positive, the y component is negative.
Meaning, you plug in only negative or positive values in this formula, then you're going to say 360 minus the theta. As simple as that. So now, in this case, since we have identified that the angle is going to be in second quadrant, since the x component of a is negative, the y component is positive.
So we're going to say, theta is going to be equal to tan inverse, AY divided by AX. Let's plug in the values. So theta is going to be tan inverse.
The AY, which we have already here, the Y component of vector A is 40 divided by 25. Okay. So our theta is going to be shift tan, open brackets, 40 divided by 25. So it's giving me 57. 0.999 which is the same as just 58.0 okay 58 degrees now we need to say 180 minus 58 degrees which is going to be 122 so in this case the direction for vector a is 122 degrees from positive x axis okay so that is it for this question So the next question is saying, express the following angles in radians. So we have 20 degrees, so we have to convert this into radian.
Now what we have to understand is that if I've got 2 pi lat, this is the same as 360 degrees. Okay. So we start from here.
In one revolution, we have got 360 degrees. Then we have 2 pi large. Now, for now, we are only interested in this. Okay.
So if I'm trying to convert the large into second, and here I can divide both sides by what? By 2, even here by 2. I can see that in one large, we're going to have 180 degrees. Meaning that what I'm going to do here is, if I say pi, not 1 large, but I have pi lad okay so in this case we can see that if i've got 20 degrees i want to convert this into what into um i want to convert this wing into what into light so meaning that i'm going to say times um i'm going to say in 180 degrees how many lads are there we've got pilot okay So 0 and 0 go.
So we're going to have 2 over 18. So we're going to have 2 there is 1. Then 2 there 9. So we've got pi over 9. Okay. So this is going to be our 20 degrees in radian. Okay. Now let's use the same method converting 50 and 100. So this 50 is going to be 50 degrees times. I'm going to have in 180 degrees there is a pi.
So this 0 and 0 will go. We're going to have 5 pi over 18. That is it. 100 degrees is going to be times 180 divided by the opposite.
It's supposed to be the opposite. We're going to have pi on top, then 180 here. So we can cancel the 0. we're going to have 10 pi divided by 18. So we can see that two into is going to be five to their nine. So I've got five over nine pi. Okay, now, the next question is same, convert the following angles to degrees.
So have got ledger to degree. So we are going to say 0.330 LAT this is going to be times okay we're going to have times we're going to say In 180, how many lad do we have? We have got pi, okay?
Pi lad. So what we're going to do, the lad and lad will cancel, okay? If the lad and lad will cancel, we're going to say 0.330. If you're using a calculator, you can find the exact value, times 180. Then the answer we're going to find, divide it by pi. okay so in this case i'm finding 18.9 which is the same as just 19 degrees okay now the next one is a 2.10 lad so i'm going to say times in we have got pilot has to be equal to 180 okay so The LAD and LAD will cancel.
We're going to have 2.10 times 180. I divide it by pi. So I'm getting 120.3. 120.32, that is going to be our angle in degrees. The next one is 7.70.
So we have a 7.70 rad. So we're going to say times in one lad which is pi, lad, there's 180 degrees. So lad and lad will cancel. We're going to have 7.70 times 180. Then I divide it by the pi.
So this is going to give us Let's redo it. We are saying 7.70 times 180. So we divide it by pi. So this is giving me 441.18 degrees. Now this is the angle we should have in this case. So that is it for this question.
So we have another question which is saying a ship set out to sail to a point 120 kilometers due north. An expected storm blows the ship to a point 100 kilometers due east of its starting point. How far, part A, how far and part B, in what direction must it now sail to reach its original destination? So it is very very important for us to understand this question. Let's come up with a free body diagram So we have the xy plane.
This ship is starting from here going to uh toward north So this is going to be our direction from this original going this direction Okay, so they are saying that in in this direction. This is going to be 120 kilometers then starting from the original again going toward east okay going toward the east so they want us to find this is 100 kilometers now they want us to find the direction and the in short we're finding the the resultant so from this point all the way to this point that is going to be our resultant okay so this is the same as we can put 120 in this direction we say we have it here That is the resultant we are trying to find. We have 100 kilometers.
We have 120 kilometers. To find the resultant in this case is going to be the square root of 100 squared. So 100 squared plus 120 squared. So this is going to give us, if we punch using our calculator, we are going to get 156.2. point two kilometers so this is the magnitude now to find the direction we know that it's going to be theta is equal to tan inverse so um in this case tan inverse we're going to have what we're going to have 120 divided by 100 our direction in this case the theta is going to be 50.2 degrees so the direction how far it is 156.2 kilometers at an angle of 50.2 degrees from positive x-axis okay now we have the next one which is same in the figure below a half piece of machinery is laced by sliding it at a distance d 12.5 meters along along a plank oriented at an angle of 20 degrees to the horizontal.
How far is it moved vertically and horizontally? So in short, in this case, they're just asking about the X and the Y component. So how far vertically they're asking about what? They're asking about the Y component. So it's VY that is vertically.
So it's going to be V sine theta. So Vy in this case, oh, we have got a D. Let's use D and not V. It's not the velocity.
It's the D. So it's going to be Dy. It's going to be D sine theta. So we're going to have our Dy, which is going D is just basically 12.5 sine.
The angle is 20 degrees. So how far vertically? is going to be 12.5 sine 20 which is going to be 4.28 so 4.28 meters that is going to be how far vertically it was moved now to find how far it was moved horizontally that is going to be the x component so we're going to have dx would be equal to d cos theta dx will be equal to d in this case is 12.5 because the theta is a 20 degrees so dx will be equal to 12 okay so what will be 12 if we do 12 cosine which is 12.5 12.5 equals 20. which is 11.75 11.75 meters that is going to be how how far it was moved horizontally okay so that is it for this question number seven a person walks in the following part 3.1 kilometers north then 4.4 kilometers west and finally 5.2 kilometers south but a sketch the vector diagram that will represent this motion b how far and c in what direction would a bed frame in a straight line from the same starting point to the same uh to the same final point So sketching vector is very very simple. Sketching is totally different from free body diagram. A free body diagram is a diagram which you yourself who is writing that diagram can understand.
But sketching we need to get exactly the information which have been given here then we represent it. Okay, now what you're going to do here is the first point what you have to do is you're going to have the starting point so the starting point let's say this is our starting point here i'm going to start from that point now this is going to be my starting point so they're saying that a person walks in the following pattern 3.1 kilometers north meaning that from this point it's going to be going in this direction 3.1 3.5 okay that is going to be 3.5 so i'll even put here to say 3.1 kilometers next from this point now another one is 2.4 kilometers west west is in this direction so 2.5 supposed to be bit shorter than the one we should have okay so I'm going to say this is going to be my my 2.2 my 2.4 kilometers so even this one I can put it outside I'm going to put it outside to say this is 3.1 kilometers. Now from this point here they are saying that finally 5.2 kilometers south.
So this is going to be 5 but this is supposed to be a bit longer than 2.4 at the same time than 3.1. So it's going to start from this point it goes there. So this is going to be our 5. so we are saying this is going to be our 5.2 kilometers so the straight line is that is going to be the result and the straight line from the starting point all the way to the last point that is the resultant so i'm going to call that one as the resultant so now the waters this is which they want so initially this is the sketching meaning we are done with part 2 we are done with part 1 which is part a but b is saying how far so it is connected to part c how far and in what direction so in this case we need first to resolve each vector into x and y component okay so let's represent this one to be vector a this one to be vector b to make things simple this one to be vector c meaning that i need to add i'm going to say i'm going to say ax plus bx plus cx plus c we only have c has to be equal to the result and now the x component of the resultant why am i adding the x component you need first to resolve each vector into x and y component add the x component alone and the y components alone so in this case we're going to have ax is going to be a cos theta plus b is going to be b cos theta c is going to be c cos theta this is going to be equal to we want to find the x component of the resultant so in this case is 3.1 okay because the angle in this case since it's going toward north the angle is 90 degrees okay plus b is 2.4 so i'm going to have 2.4 because the angle in this case since it's going toward negative x-axis which is west the angle is 1.8 plus c is 5.2 because the angle since it's going toward south is 2.7 has to be equal to the x component resultant so we're going to have um 3.3.1 cos 90 plus 2.4 cos 180 plus 5.2 cos 270 so this is going to give me negative 2.4 so this is giving me negative 2.4 is equal to the x component of the resultant now that i have the x component let's find the y component so the rx is negative 2.4 okay kilometers so i'm going to get rid of this and now i need to find the y component so what i'm going to do here i'm going to say ay plus by plus CY is equal to the Y component of the resultant.
So it's going to be this in Y. Y component is going to be sine. Sine theta plus B sine theta plus C.
Sine theta has to be equal to the ROI. Now A is 3.1. This sine 90. The angle is 90 degrees plus B is 2.4.
Sine E. angle in that case is 180 degrees plus this is 5.2 sine 270 has to be equal to the y component so in this case we're going to have 3.1 sine 90 plus 2.4 sine 180 plus 5.2 sine 270 so the answer i'm getting is negative 2.1 that is my y component so the y component of the resultant is a negative 2.1 kilometers now to find the magnitude of this resultant this is what we are going to do we are going to say the resultant is going to be the square root of rx squared plus ry squared so that is the displacement which you're finding which is alpha so we are going to have the square root of negative 2.4 squared plus negative 2.1 squared so we're going to have 2.4 squared plus 2.1 squared now when you're plugging in the values here using a calculator you just need to know the negative because it is squared so the negative is going to be cancelled okay so we are going to have 3.8 which is the same as 3.2 kilometers now that we have the resultant we need to find the direction so it is 3.8 3.2 kilometers now in what direction to find the direction us knowing that the the x component is negative the y component is negative we expect our angle to be in third quadrant leave it from the diagram here we can see that this is going to be in the third quadrant Okay, you can see it's in third quadrant. So we need to see so from here We can see from the x and y component that the angle is going to be in third quadrant So if the angle is going to be in third quadrant, we're going to say the answer which we're going to find We're going to add the theta with our t1 h.
So it's going to be theta is equal to tan inverse RY divided by RX so theta will be equal to tan inverse RY is we're going to give the negative is 2.1 then this is 2.4 so we have shift tan open brackets 2.1 divided by 2.4 so this is giving me theta to be equal to 41.2 degrees so now since it's supposed to be in third quadrant we are going to say 180 plus 41.2 degrees so the direction in this case is going to be you it's going to be 180 180 plus 41.2 which is going to be 200 and 221.2 degrees that is going to be the direction so that is it for this question So our question 8 is saying you are to make 4 straight lines moving over your flat desert floor starting at the origin of an x y coordinate system and ending at the x y coordinate which is negative 140 meters comma 30 meters. So this is our resultant in this case. So these are the coordinates for the resultant meaning that the x component because the first part is the x. x component meaning that the x component of the resultant is 114 meters the y component of the resultant in this case is going to be 30 meters so they're saying that in the x component and the y component of your move are the following uh respectively in meters so the first part we can say it is a you can just assume that that is a meaning we have got ax which is 20 and we have got our bx to be equal to what? Oh, our Ay, sorry.
Our Ay to be equal to 60. Then you're going to have B, which is going to be Bx, the one we don't know. The By is negative 70. Now let's go to C. Our Cx is negative 20. Then our Cy is, we don't know about Cy. Let's have also, we say we have also D. So Dx is negative 60. Then our Dy is...
negative 70 so it's just a matter of us adding them so to find the first question is saying what are the components what are the components B and the component C then but see the last question saying what are the magnitude and the election which is the angle relative to the positive direction of the x-axis of the overall displacement. Now to find the x and y component, the one which are missing in this case, we're going to say ax plus bx plus cx plus dx has to be equal to the resultant which is the x. Okay, so what is AX? AX in this case is 20 plus what is our BX?
Our BX is the one we're trying to find. What is our CX? Our CX in this case is negative 20. What is our DX? Our DX in this case is our DX is negative 60. Then the resultant now the X component is 140. So now we're going to say what is 20 plus which is minus we can just put minus 20 again minus 60 so it's negative 60 so we have got negative 60 plus bx is equal to negative 140 so i want to solve for bx is going to be negative 140 plus 60 so meaning that the bx which is the x component of that one is going to be um we're going to have 140 minus that one we're going to get a negative 80 negative 140 plus 60 is negative 80 meters so that is the component of b to find that other one for y now we're going to use the one for y since we want to find the one which is missing which is c so we're going to have our our a y plus b y plus c y plus d y is equal to r y so a y is 60 plus b y is negative 70 c y is the one we're trying to find then d y is negative 17 then the r r y which is the resultant is 30 so we're going to say 60 minus 70 minus 70 which is going to be negative 80. so it's going to be negative 80 plus cy is equal to 30. so cy will be equal to shift this negative 80 to the other side to be 30 plus 80. So in this case, our CY is going to be 30 plus 80 is 110 meters. So that is the CY.
Now, the last question is saying we need to find the magnitude and direction of the resultant. The resultant, we have the X and the Y components. So it's going to be the resultant, the square root of Rx plus Ry.
So we square them. So our resultant in this case is going to be, we have negative 140 squared plus 30 squared. So our resultant in this case, we expect to have 143.2 kilometers.
What of the direction? Now, before you find the direction, first consider the x and y component. That's why the question is saying, It's supposed to start from positive x-axis relative to the positive direction of x-axis.
So the x is negative, the y is positive. We expect our angle to be in second quadrant. Okay.
Now in second quadrant, what do we do? 180 minus the theta which we're going to find. So we know that the theta is going to be tan inverse. The Ry divided by Rx then this is going to be theta will be equal to We're going to have done in this we have our Ry to be 30 I'm going to ignore the negative because I'm using this formula if I include the negative then I'm going to say 180 plus Vita Because this fit is going to be negative Okay, but no one is going to penalize you even if you ignore the negative so it's going to be 140 So I'll get my theta to be 12.1 degrees. Now I have to say 180. I'm going to say 180 minus this.
Okay. So we're going to have 180 minus 12.1 degrees. So this is going to give me from positive x-axis.
This is going to give us 168. It's 167.9. 167.9 degrees but we can just launch it off and say it's going to be the fit is going to be the direction is going to be 167.9 so it's going to be 168 degrees from positive x axis so that is it for question 8 number 9 if b is added to c which c is equal to 3.0 i plus 4.0 j the result is you The result is a vector in positive direction of the y-axis with the magnitude equal to that of c. What is the magnitude of b? So they are saying that if c is added to b, so if we say b plus c, this is giving us the magnitude of c. But they are saying that it's going toward positive y-axis, meaning that the angle is 90 degrees.
Now in this case, what we are going to do is we are going to have two equations. We know that this B, we are going to have this B plus C is the same as this. Before we even come here, let's say our B, let's say our vector B is X, I plus Y, J. Meaning that if we add the I alone, we're going to have 3 plus X, then we have got I plus, we're going to have 4 plus Y, we have J. Okay.
But at the same time, we know that if this is equal to this. Then we can put this one to be the resultant to say our b plus c, this one is going to be the square root of 3 plus i, then squared, 3 plus x squared, which is basically i plus 4 plus y squared, which is j. Now we know that what we have here. There is a square and square.
We can just get what is inside. So in this case, we're going to say the resultant we've been given already. Okay.
So to find the resultant, we know that the resultant is the same as that of C. Meaning that to get the resultant is going to be the square root of 3 squared plus 4 squared, which is going to give us the resultant is going to be 5. Okay, now this 5, since we're saying that the angle is 90, meaning that the x which is the i, in this case is going to be r, okay, is going to be rx, which is r, rx is same as ri, okay, ri is going to be r cos theta, ri is going to be, r is 5. cos 90, which is going to give us zero. Then, if we are going to do the same, we're going to say RI, which is going to be Rj now, so it's going to be R sine theta.
Now in this case, we're going to have j, R is five, we're going to have sine 90. So Rj is going to be five. Now we can see that we have we can compare this and this, okay. We can compare this and the x component which is this one, 0. So we are going to get this which is inside here, 3 plus x, okay, which is i, is going to be equal to ri. So we are going just to get 3 plus x will be equal to, the answer is 0. In this case, x will be equal to negative 3, meaning we are done with x. Okay, to get the y, what we're going to do is we're going to compare again 4 plus y will be equal to the 5. This now, okay, meaning that y will be equal to 5 minus 4, y will be equal to 1. Now, our b, remember, our b we found that b was xi, okay, b was xi plus yi.
So BX is what? Our X in this case is negative 3, I plus our Y is 1, which is just basically I, which is J. Okay, it's going to be, this is J and not I, so it's going to be J. so in this case this is the magnitude of what the b okay so this is going to be what the b but if we want to get the magnitude of this b we know that it's going to be b is going to be equal to negative 3 squared plus negative 1 squared so what would there be in this case the magnitude of b is going to be um 3 squared plus 1 squared then we get the square root so the square root of 10 which is going to be 3.2 so we can say that is 3.2 that is going to be the magnitude but you can even leave your answer here to say b is so B is a negative 3i minus j. That is just okay.
Or you can find this. After finding this, you also get this. Then you're going to get the magnitude to be 3.2.
As simple as that. Here is our question. The question is saying, in the figure below, a vector A with a magnitude of 17 meters is directed at an angle of 56 degrees counterclockwise from positive x-axis.
What are the components, a, aX and aY of the vector? Then they are saying a second coordinate system is inclined by an angle of 18 degrees with respect to the first one. What are the components, x and y, in this primed coordinate system?
This question is asking us to find the x component and the y component. In this case, what we're going to do is we're going to say we have got A, so it's going to be AX, it's going to be A cos theta. Okay, we have been told that this angle, which is 56 degrees, is counterclockwise. So counterclockwise meaning it is coming from positive X axis. And for sure, whenever we're talking about vectors, we are measuring the angles from positive X axis until it touches the vector.
Okay, so we're going to use the same angle 56. So we're going to say a x is going to be the vector which we've been given is 17 cos the theta is 56. So the x component in this case is going to be the x component in this case is going to be 17 cos 56, which is 9.5. So 9.5 meters is our X component. So I'm going to put it here. 9.5 meters is our X component. Let's now go ahead and find the Y component.
So the Y component in this case is going to be AY is going to be A sine theta. So AY is going to be, what is our A? Our A is 17. We have sine 56. So what is 17 sin 56? So this is giving me 14.09, which is the same as 14.1 meters. So in this case, our Y component is 14.1 meters.
Now, the second part of this question is saying, a second coordinate system is inclined by an angle of 18 degrees with respect. to the first one so as we can see the second angle is now here see the angle they're talking about but remember from this point all the way to this vector it is 18 it is 56 degrees so to get the angle to get this angle only we're going to say the whole angle here is what 56 so i'm going to say 56 minus 18 okay Meaning that that is going to be the angle which was being inclined there. So we are going to see that this is going to give us 56 minus 18. It is 38. So we're going to have our angle 38. So this is the angle which we're going to be using. The magnitude is the same.
So we're going to say Ay, Ax now inclined, which is, there is a prime. It's going to be A prime cos theta. So we're going to have, we're going to have 17. then you have cosine 38 so a prime x is going to be 17 cosine 17 cos 38 is giving me 13.396 which is the same as 13.4.
So this is going to be, so we have managed to find 1. Let's now go ahead and find the y component. So it's going to be a'y, it's going to be a'sin theta. So we are going to have, we are going to have A in this case is 17, okay, sine 38. So A prime Y is going to be 17 sine 38 is going to be 10.46, which is the same as 10.5 meters. So these are the X and Y. components so these are the answers for this question okay