Adding Vectors Graphically
Adding Vectors Graphically
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Hey, guys. So previously we saw that vectors air just triangles and vector math turns into a bunch of triangle math. Well, you're gonna need to know how to add vectors a lot in physics. So in this video, I'm gonna show you how to add vectors. Graphically, I feel like it's a great way to visualize what's actually happening. When you are combining multiple vectors, let's check it out. So remember that vectors are drawn as arrows, like a displacement vector or something like that. And the way that we add vectors is we just connect those arrows and we're gonna do this in a way that your textbooks and professors called tip to tail. We actually saw this when we added perpendicular vectors. Let's check it out. If you were to walk 3 m to the right and then 4 m up, all we did was we basically just connected these arrows tip to tail, and your total displacement was actually is if I had actually just walked from here to here. We called this C and the way that we calculated this was we found out that these arrows just made a triangle. We could use the Pythagorean theorem. So this magnitude here was the square root of three squared plus four square, and that was 5 m. So this total displacement that we found, which is just the shortest path, gets a fancy name. It's called the resultant vector Sometimes or the result in displacement. Sometimes you'll see, see or are for that resultant. And basically the resultant is always going to be the shortest path from the start of the first to the end of the last. So just like we did over here, it's as if you had actually walked in this direction. So one thing that we haven't seen yet is how to add vectors that aren't perpendicular. And so we've got these two vectors here. And to add them, we just have to use the tip to tail method. We just have to add and connect these things tip to tail, but notice that they both start at the same place. So one thing I can do here is Aiken, basically pretend is if I can pull this vector if I pull this thing over to the right like this so that I can connect them eventually, I wanna line up so that the eso that can connect these vectors tip to tail And what you would get is something that looks like this. We know that vector A is gonna be two to the right and one up. So that means my vector is gonna be two to the right and one up that looks like this. And then my be vector is gonna be one to the right and three up. So from the end point of here, I'm gonna go one to the right and three up and so I'm gonna end up over here. So these air the vector's connected tip to tail. And so the result in vector, which is the total displacement, is the shortest path from start to finish. So here is the, um, the start of the first one and the end of the last one. And so this is my displacement vector here. So this is my resultant And so the way I calculate this, the magnitude is I just break it up into a triangle, which I know the legs of this triangle are gonna be these legs right over here, and I can get the numbers just by adding up the boxes and counting up the boxes. I've got three here and four here. So that means that the magnitude is three squared four squared, which is equal to 5 m. So this is how I add together A and B. What if instead I want to add B plus A. So basically, I'm just going to do the opposite. I'm going to start off with the B vector first, which I know is one to the right and three up. So that means it's gonna go like this. So this is my B vector, and then my a vector is to to the right and one up. So that means from here I'm gonna go to to the right and one up. So connect them tip to tail and the displacement or the resultant is gonna be the shortest path from start to the end. So that means this is my displacement. Over here, it's c break it up into a triangle and get the same exact legs that I did before. I've got three and four. And so the Pythagorean theorem, the three squared plus four squared square rooted is going to be 5 m. So notice here that it didn't matter the way that we added the vectors together a or a plus B or B plus A. We ended up at the same exact point. We ended up with the same exact arrow. So that means that adding vectors does not depend on the order. This is something that your textbooks call. The cumulative property just means that three plus two is five. Two plus three is also five. So that just means that it doesn't matter the way that you add vectors up together. You always just gonna get the same number. Alright, guys, that's it for this one. Let's go ahead and get another example. We're gonna find the magnitude of this result in vector A plus B. So we're just gonna basically combine these vectors so that they are tip to tail because they're not right now. And then we just find out the shortest path between start and finish. So I'm gonna add a plus B. So I'm gonna start off with a and then B here is gonna be one to the left and four down. So that means that from the end of be, I'm gonna go one to the left and four down. So +1234 So that means that might be vector Looks like this. So I kind of just imagine that I just pulled this be vector about this way to the right there. So they line up tip to tail, so I kind of just pretend that this isn't there anymore. So now what's the resulting vector? Well, the start of the first one was here, and the end of the last one is over here. So that means that my displacement vector is the shortest path. And that's the arrow right here. So this is my displacement vector. Weaken. Break it up into a triangle. Basically, these are gonna be the legs of this triangle, and I can count up the boxes. I've got three here and three here, so that means the magnitude of the sea vector, which is usually written by a bunch of vertical bars, um, is just gonna be the Pythagorean theorem. So you got three squared and three squared, and what you're gonna get is 4.2 m. Alright, guys, that's all there is to it. Let me know if you have any questions.
A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the +x-direction. What is the magnitude (in miles) of its final displacement from the origin?
Adding 3 Vectors
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Hey, guys, let's work this one out together. We're gonna find the magnitude of a resultant vector by adding up these three vectors over here. So let's check it out. So we've got these three vectors, but they all actually start from the same place, the origin. Which means we're gonna have to connect them tip to tail, that we're gonna find the shortest path from start the first to the end of the last one. That's what always we do for Vector Edition. So let's check it out. So we have these the a vector that's already drawn from the origin. And so all have to do is just connect the B vector tip to tail. So I've got this be vector here. But if you think about this, this be vector goes three to the left and then it goes five up. So the one way I can sort of move it over is start from the end of a and I could just go three the left and then five up. And then my results in our sorry the B vector is just gonna be right here. So basically, I'm kind of like transplanting it over. So that it lines up tip to tail. So this is my B vector. Kind of just erases because I don't need it anymore. Now, my see, vector is gonna be the same thing. It's gonna be five to the left and two down. You're just gonna add it now? Tip to tail from this point over here. So you go. 12345 and then you go. 12 you're gonna end up over here. And so now this is going to be my see Vector added tip to tail. So this is C, which means I don't need it anymore. Notice how all these things are also parallel. So this PC vector is parallel. D. C vectors are parallel. That just means that you drew them in the right way. So that means now that my result in vector is gonna be the start of the first to the end of the last. Actually, instead of you walking this whole path from A to B to C, it is actually it just if you walked in a straight line that connects these two points So this is my d vector here. Now I just have to figure out the magnitude. So I'm just gonna break it down into a triangle, figure out the legs, just point counting up the squares. So I've got this triangle over here and I've got 123456 And then 123456 So I've got a six and six triangle. So now, to calculate the magnitude, you just use the Pythagorean theorem and you have to worry about the signs. Just plug in the numbers of the boxes. Six squared plus six squared and you just get 8.5. And if you wanted, if it was meters or something like that, then you just specify the units. So that's the resultant or that is the magnitude of that. Resultant. Let me know if you guys have any questions.
Subtracting Vectors Graphically
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Hey, guys. So now that we've seen how to add vectors together and some problems, you'll have to subtract them. So in this video, I'm gonna show you how to subtract vectors. Graphically, What we're going to see is that it's exactly like how we added vectors. You're gonna combine a bunch of arrows tipped the tail. The only thing that's different here is that one or more of the vectors is going to get reversed. Let's check it out. So when we added vectors, just a quick recap, you would connect them tip to tail like this, a plus B. And the resultant vector was the shortest path from start to finish. The start was here and the end of the last ones here, and this was my result in Vector. We called it C, and basically we just conform a little triangle like this. And then we would counter the boxes to get the legs three and four on that account to get the magnitude of the total displacement or the resultant. We just use the Pythagorean theory for three squared four squared and that was five. Now, if we do B plus A, we're gonna do the same exact thing. The only thing that's different is that the vectors get reversed. Uh, sorry that the vectors get added in reverse, so we just do B plus A. But the shortest path is still from start to finish like this, and then we end up with the same exact triangle because we're gonna end up with three and four. So what if instead of a plus B, I want to do a minus bi Now, guys, the big difference here. The key difference is that when you're doing vector subtraction, one way you can think about this is we're just doing a plus the negative of be so vector subtraction is really just vector addition. It's just that one of the vectors has a negative sign. So how does that work? Well, let's check it out. So the a vector in our example was one or to the right and one up. So if you want to add these things, we have to connect them tip to tail. So from the origin, I'm gonna go to the right and one up like this. So this is my a vector. And then if I want to add a and B, I would have to go tip to tail and might be vector pointed one to the right and three up. So I'm gonna go one to the right and three up, and it would look like this. So this is my B vector over here, but I'm not trying to add a and B I'm trying to add a negative. Be. So what happens? What's the deal with that negative sign? Well, negative signs in physics just have to do with direction. So if positive B is one to the right and three up, then the negative of B is just gonna be If I flip those two things and I go one to the left and three down, So by negative be vector is gonna point in this direction over here. Basically exactly opposite. So notice how these two things have the same length, but they're pointing in perfectly opposite signs are perfectly opposite directions. So the negative of a vector is gonna have the same magnitude, but it's gonna points in the opposite direction. So let's go ahead and add them now. So now we're not gonna add these two vectors together. We're gonna add these two vectors together. The result is still gonna be the shortest path from start to finish the starts here and the end of last ones here. So that my shortest path, it's just the straight line that connects those things. So basically, it's if I'd actually just walked in this direction here so we couldn't make the little triangle weaken Count of the boxes for the legs one and two and the high pot news or the magnitude is gonna be one squared plus two squared. And that's 2.24 now. What if I wanted to do B minus a and basically reverse the order like I did before? Well, we can think about this using the same exact principle. B minus A is the same thing as we're one week we can think about. This is B plus the negative of a So now we're gonna do the same exact thing. We're just gonna start off with the vector First, my be vector points one to the right and three up. So it's gonna look like this. And if I if I wanted to add being a, I connect them tip to tail and my A vector is to the right and one up. So it looks like that. So this is a vector, but this is positive. A So that means that negative A would just be pointing in the exact opposite direction. So would be to the left and one down. So my negative A just points exactly the opposite direction. Like this. And so I'm gonna add these two vectors together. So this is my B not this one. Not this positive. A. So my shortest path from start to finish is from the origin up to this point Over here, the straight line is basically as if I had just walked in that direction instead of going here and then backwards like that. So this is my c break it up into a triangle. We're gonna end up with one and two. And so when we do the high pot news, we're gonna use the same exact numbers when you get one squared in two squared, so the magnitude is going to be 2.24 What is different, though, is the direction. So let me summarize. We did Vector edition. The order didn't matter whether we did A and B or B and a we ended up at the same points are displacement vectors point in the same direction we do vector subtraction. However, the order does matter, you're gonna get the same magnitude. But here these vectors point exactly in opposite directions whether you do a minus bi or B minus a. So be careful when you're adding those things on, make sure you're doing in the proper order. Alright, guys, that's all there is to it. Let's go ahead and get to an example. So we're gonna calculate the magnitude of this result in vector here. We've got a minus bi. So one way we can think about this is we're just doing a plus the negative of be So let's check it out. We've got these two vectors here, but they're actually not lined up tip to tail. They're both they're both basically starting from the same place. So we've got our A vector like this and now we have to add it tip to tail with negative of be So we're gonna start over here and now if b is gonna be to the right and for up then my negative B is just gonna be if I reverse it. So I'm gonna go to to the left and I'm gonna go 1234 down to my negative be vector. Looks like this the exact opposite direction. So this is actually what I'm adding A a negative B. Which means that my resultant is just gonna be the shortest path from start to finish is if I basically walked in this direction instead of doing here and that instead of doing both of those motions there. So this is my new displacement vector. My resultant. And so it break it up into the triangle kind of the boxes. I've got four in this direction and to in this direction. So the magnitude, which is what I'm looking for here, it's just gonna be the Pythagorean theorem. So I've got four squared and two square Don't have to worry about, you know, the boxes points in the left or anything like that. So you just add up the numbers and you're gonna get four points 47. So let's call that meters. Alright, guys, that's all there is to it. Let me know if you have any questions
Find the magnitude of the Resultant Vector D=C-B-A.
Adding Multiples of Vectors
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Hey, guys. So up until now we've seen how to simply adventures together like A and B. But sometimes in problems, you're gonna have to add multiples of vectors, like to a plus three B or to a plus 0.5 be something like that. What we're gonna see is that it works exactly like Vector Edition. The only thing that's different is that when you multiply a vector by sticking the number in front of it like a becomes to a, what's happening is because they have a number in front of it. The magnitude of the length is gonna change, but not the direction. Let's check it out. So when we had vectors A and B, basically you would just line them up, tip to tail like this, and your result in vector is the shortest path from the start of the first to the and the last. It's basically is if you had walked in this direction, if you were like calculating the total displacement or something like that and so we just break this up into a triangle and then we count up the boxes to figure out the legs. This is five and five And so the magnitude would just be five squared and five squared. And that's 7.7 What if instead of a plus B, I was given to a plus 0.5 B. Well, one way you can think about this. These multiples air these numbers in front is you can think about this to a here as just being a plus A. So you're still just doing vector addition here. This number here is basically kind of just like condensing all of this information just into a single number. And these numbers are going to change the length of the vectors. So this is a plus a and then you're gonna add it to 05 B, which you can think of is like half of a B. Let's check it out. So if my vector A is three to the rights and one up, then that means that a plus a or to A is just gonna be If I have three of the right and one up like this, and then you just add a tip to tail to another 13 to the right and one up like this, so we're gonna have a and a and basically this whole entire vector here is to a so that if I had added 2.5 b, we do the same thing. So my 0.5 b or sorry, my regular be vector is to to the right and four up. So that means that half of bees, if I just cut everything in half. So instead of to the right and four up, I'm just gonna go one to the rights and to up like this. So this is gonna be my 10.5 b vector. And so my results in vector is just gonna be the shortest path from the start to the end. So this is going to be the result in vector here. So what we can see is that whenever you when you multiply a vector by a number that's greater than one, what you're gonna dio is you're gonna basically increase the magnitude or the length. So, for instance, this a here points in the same exact direction as this A. It's just twice as long. And then when you multiply a vector by number, that's less than one like we did for this one. It points in the same exact direction. It's just gonna be decreased in terms of the magnitude and the length. So what happens is and number that's greater than one makes the vector longer less than one makes the vector shorter. That's really all there is to it. So we could just calculate the resulting vector by you know, this is my see, we just count up the legs over here. And this is 1234567 This is 1234 So my hi pop news seven square post four squared and you get 8.6 Alright, guys, that's all there is to it. Let's go ahead and get another example. So we're gonna find the magnitude of this result in vectors, see? And it's gonna be 38 minus to be. So, by the way, all these rules work even for vector subtraction. Let's check it out. So we've got these two vectors here A and B, and basically all I have to do is we can think of this. Three A. Here is just being a plus a plus a just three vectors stacked on top of each other, lined up tip to tail and then this to be here. We're going to subtract B plus B. So let's go ahead and stack all those vectors together. We know that A is gonna be one to the right and one up, So that means three is just gonna be if I stack tomb or on top of those. So this whole entire thing here ends up being three a And so now we just have to add the negative to be well, my regular be vector is gonna be three to the right and one up. So from the tail of this one from the end of this one, I have to go. Not three the right and one up. I have to go in the opposite direction. I have to go through the left and then one down. So this is gonna be my negative be vector. So this is my negative B and I have to do it again. I have to do another three the left and then one down. So I'm gonna go this way. So notice how we've got this be vector here and now this vector is exactly twice as long. But now it just points in the opposite direction. That's all. That minus sign does. So now our result in vector here is gonna be from the start of the first to the end of the last one is basically if I just walked in this path, um and so you're gonna just draw the shortest path between those. This is my see vector, and then you just count up the legs. This is three and this is one. And so the magnitude which is given by the absolute value sign, is the square root of three squared plus one squared and you just get 3.16 and that is the magnitude. Alright, guys, that's all there is to it. Let me know if you have any questions.
Additional resources for Adding Vectors Graphically
PRACTICE PROBLEMS AND ACTIVITIES (3)
- A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east...
- For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of (b) the vec...
- For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of (a) the vec...