3. Vectors

Adding Vectors Graphically

# Adding Vectors Graphically

Patrick Ford

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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.

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