Adding Vectors Graphically - Online Tutor, Practice Problems & Exam Prep

Vectors can be added graphically using the tip-to-tail method, where the resultant vector represents the shortest path from the start to the end point. The magnitude of the resultant can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$. When subtracting vectors, one vector is reversed, and the same principles apply. Multiplying vectors by scalars changes their magnitude but not their direction, affecting calculations of resultant vectors in physics.

Adding Vectors Graphically

Hey, guys. So previously, we saw that vectors are just triangles, and vector math turns into a bunch of triangle math. Well, you're going to need to know how to add vectors a lot in physics. So in this video, I'm going to 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 going to do this in a way that your textbooks and professors call tip to tail. We actually saw this when we added perpendicular vectors. Let's check it out.

If you were to walk 3 meters to the right and then 4 meters up, all we did was we basically just connected these arrows tip to tail, and your total displacement was actually as if I had actually just walked from here to here. We called this c, and the way that we calculated this was we found that these arrows just made a triangle. We could use the Pythagorean theorem. So this magnitude here was the square root of 3 squared plus 4 squared, and that was 5 meters. 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 resultant displacement. Sometimes you'll see c or r 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, just like we did over here. It's as if you had 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 2 vectors here, and to add them, we just have to use the tip to tail method. We just have to add or connect these things tip to tail. But notice that they both start at the same place. So one thing I can do here is I can basically pretend as 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 want to line up so that I can connect these vectors tip to tail. And what you would get is something that looks like this. We know that vector a is going to be 2 to the right and one up. So that means my vector a is going to be 2 to the right and one up. It looks like this. And then my b vector is going to be 1 to the right and 3 up. So from the endpoint of here, I'm going to go 1 to the right and 3 up, and so I'm going to end up over here. So these are the vectors connected tip to tail.

And so the resultant vector, which is the total displacement, is the shortest path from start to finish. So here's 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 going to be these legs right over here. And I can get the numbers just by adding up the boxes and counting up the boxes. So I've got 3 here and 4 here. So that means that the magnitude is 32 + 42, which is equal to 5 meters.

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 1 to the right and 3 up. So that means it's going to go like this. So this is my b vector. And then my a vector is 2 to the right and 1 up. So that means from here, I'm going to go 2 to the right and 1 up. So connect them tip to tail, and the displacement or the resultant is going to be the shortest path from the start to the end. So that means this is my displacement over here. It's c. I break it up into a triangle and get the same exact legs that I did before. I've got 3 and 4, and so the Pythagorean theorem, the 32 + 42, square rooted, is going to be 5 meters.

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. It just means that 3+2 is 5, 2+3 is also 5. So that just means that it doesn't matter the way that you add vectors up together. You're always just going to get the same number.

Alright, guys. That's it for this one. Let's go ahead and get another example. We're going to find the magnitude of this resultant vector, a plus b. So we're just going to 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 going to add a plus b. So I'm going to start off with a. And then b here is going to be 1 to the left and 4 down. So that means that from the end of b, I'm going to go 1 to the left and 4 down. So 1, 2, 3, 4. So that means that my b vector looks like this. So kind, I kind of just imagine that I've just pulled this b vector about this way to the right there so that they line up tip to tail. So I can kind of just pretend that this isn't there anymore. So now what's the resultant 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. We can break it up into a triangle. Basically, these are going to be the legs of this triangle, and I can count up the boxes. I've got 3 here and 3 here. So that means the magnitude of the c vector, which is usually written by a b

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?

• A. 17 miles
• B. 13 miles
• C. 9,0 miles
• D. 6.4 miles

Adding 3 Vectors

Hey, guys. Let's work this one out together. We're gonna find the magnitude of a resultant vector by adding up these 3 vectors over here. So let's check it out. We've got these 3 vectors, but they all actually start from the same place, the origin, which means we're going to have to connect them tip to tail. Then we're going to find the shortest path from the start of the first to the end of the last one. That's what we always do for vector addition. So let's check it out. We have the a vector that's already drawn from the origin and so all I have to do is just connect the b vector tip to tail.

I've got this b vector here, but if you think about this, this b vector goes 3 to the left and then it goes 5 up. So the one way I can sort of move it over is to start from the end of a and I can just go 3 to the left and then 5 up and then my result in or sorry, the b vector is just going to 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 and I kinda just erase this because I don't need it anymore.

Now, my c vector is going to be the same thing or it's going to be 5 to the left and 2 down. You're just going to add it now tip to tail from this point over here. So you go 1, 2, 3, 4, 5, and then you go 1, 2, so you're going to end up over here. And so now, this is going to be my c 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 c vector is parallel. These c vectors are parallel. That just means that you drew them in the right way.

So that means now that my resultant vector is going to be the start of the first to the end of the last. It's actually instead of you walking this whole path from a to b to c, it is actually just if you walked in the 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 going to break it down into a triangle, figure out the legs just by counting up the squares. So I've got this triangle over here and then I've got 1, 2, 3, 4, 5, 6 and then 1, 2, 3, 4, 5, 6. So I've got a 6 and 6 triangle.

So now, to calculate the magnitude, you just use the Pythagorean theorem. You don't have to worry about the signs, just plug in the numbers of the boxes, 6 squared plus 6 squared, and you just get 8.5. And if you wanted, if it was meters or something like that, then you would 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.

Here, the calculation of the magnitude may be incorrect due to rounding or calculation errors. Correctly, if given for a 6 by 6 sided right triangle: 6 2 + 6 2 But please verify with correct calculations or tools.

Subtracting Vectors Graphically

Hey, guys. So now that we've seen how to add vectors together, in some problems, you'll have to subtract them. In this video, I'm going to 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 going to combine a bunch of arrows, tip to 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.

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 one is here, and this was my resultant vector. We called it c, and basically, we just can form a little triangle like this, and then we would count with the boxes to get the legs 34. And then to count to get the magnitude of the total displacement or the resultant, we just use the Pythagorean theorem for 3 squared and 4 squared, and that was 5. Now if we do b+a, we're going to do the same exact thing. The only thing that's different is that the vectors get reversed. Sorry. 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 going to end with 34.

So what if instead of a+b, I want to do a minus b? 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 b. 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 2 to the right and 1 up. So if we wanted to add these things, we have to connect them tip to tail. So from the origin, I'm going to go 2 to the right and 1 up like this. So this is my a vector. And then if I want to add a and b, I'd have to go tip to tail, and my b vector pointed 1 to the right and 3 up. So I'm going to go 1 to the right and 3 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 and negative b. 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 1 to the right and 3 up, then the negative of b is just going to be if I flip those two things. And I go one to the left and 3 down. So my negative b vector is going to 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 directions. So the negative of a vector is going to have the same magnitude, but it's going to point in the opposite direction.

So let's go ahead and add them now. So now we're not going to add these 2 vectors together. We're going to add these 2 vectors together. The resultant is still going to be the shortest path from start to finish. So the start is here, and the end of the last one's here so that my shortest path is just the straight line that connects those things. So, basically, it's if I'd actually just walked in this direction here. So we can make the little triangle. We can count out the boxes for the legs, 1 and 2, and the hypotenuse, either the magnitude, is going to be 1 squared plus 2 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 b plus the negative of a. So now we're going to do the same exact thing, but we're just going to start off with the b vector first. My b vector points 1 to the right and 3 up, so it's going to look like this. And if I wanted to add b and a, I connect them tip to tail, and my a vector is 2 to the right and one up. So it looks like that. So this is my a vector. But this is positive a, so that means that negative a would just be pointing in the exact opposite direction. So it'd be 2 to the left and one down. So my negative a just points exactly in the opposite direction like this. And so I'm going to add these 2 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. It's 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 going to end up with 1 and 2. And so when we do the hypotenuse, we're going to use the same exact numbers when you get 1 squared and 2 squared, so the magnitude is going to be 2.24.

What is different though is the direction. So let me summarize. When we did vector addition, the order didn't matter. Whether we did a and b or b and a, we ended up at the same points. Our displacement vectors point in the same direction. When we do vector subtraction, however, the order does matter. You're really going to get the same magnitude, but here these vectors point exactly in opposite directions whether you do a minus b or b minus a. So be careful when you're adding those things, and make sure you're doing it 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 going to calculate the magnitude of this resultant vector here. We've got a minus b. So one way we can think about this is we're just doing a plus the negative of b. 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 starting from the same place. So we've got our a vector like this. Now we have to add it tip to tail with a negative of b. So we're going to start over here. And now if b is going to be 2 to the right and 4 up, then my negative b is just going to be if I reverse it. So I'm going to go 2 to the left, and I'm going to go 1, 2, 3, 4 down. So my negative b vector looks like this, the exact opposite direction. So this is actually what I'm adding, a and negative b, which means that my resultant is just going to be the shortest path from start to finish as 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 and my resultant. So I break it up into the triangle, count off the boxes. I've got 4 in this direction and 2 in this direction.

So the magnitude, using the Pythagorean theorem. So I've got 4 squared and 2 squared. You don't have to worry about, you know, the boxes pointing to the left or anything like that. So you just add the numbers and you're going to get 4.47. 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.

• A. 2.24
• B. 3.0
• C. 12.5
• D. 17

Adding Multiples of Vectors

Hey, guys. So up until now, we've seen how to simply add vectors together like a and b. But sometimes in problems, you're going to have to add multiples of vectors like 2a + 3b or 2a + 0.5b something like that. What we're going to see is that it works exactly like vector addition. The only thing that's different is that when you multiply a vector by sticking a number in front of it like a becomes 2a, then what's happening is because they have a number in front of it, the magnitude or the length is going to change, but not the direction. Let's check it out. So when we have vectors a and b, basically, you just line them up tip to tail like this, and your resultant vector is the shortest path from the start of the first to the end of the last. It's basically as 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 55 and so the magnitude would just be 52 + 52 and that's 7.07. What if instead of a + b, I was given 2a + 0.5b. Well, one way you can think about this, these multiples or these numbers in front is you can think about this 2a here as just being a+a. So you're still just doing vector addition here. This number here is basically kinda just like condensing all of this information, just into a single number. And these numbers are going to change the lengths of the vectors. So this is a + a and then you're going to add it to 0.5b, which you can think of as like half of a b. Let's check it out. So if my vector a is 3 to the right and one up, then that means that a + a or 2a is just going to be if I have 3 to the right and one up like this. And then you just add a tip to tail to another one, 3 to the right and one up like this. So we're going to have a and a And basically, this whole entire vector here is 2a. So then if I had to add it to 0.5b, we do the same thing. So my 0.5b Or sorry. My regular b vector is 2 to the right and 4 up. So that means that half of a b is if I just cut everything in half. So instead of 2 to the right and 4 up, I'm just going to go 1 to the right and 2 up like this. So this is going to be my 0.5b vector. And so my resultant vector is just going to 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 1, what you're going to do is you're going to 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 a number that's less than 1 like we did for this one, it points in the same exact direction. It's just going to be decreased in terms of the magnitude and the length. So what happens is a number that's greater than 1 makes the vector longer. Less than 1 makes the vector shorter. That's really all there is to it. So we can just calculate the resultant vector, by, you know, this is my c. We just count up the legs over here and this is 1, 2, 3, 4, 5, 6, 7. This is 1, 2, 3, 4. So my hypotenuse is 72 + 42 and you get 8.06. Alright guys. That's all there is to it. Let's go ahead and get another example. So we're going to find the magnitude of this resulting vector C and it's going to be 3A - 2B. So by the way, all these rules work even for vector subtraction. Let's check it out. So we've got these 2 vectors here, a and b. And basically, all I have to do is we can think of this 3a here as just being a + a + a. Just 3 vectors stacked on top of each other lined up tip to tail. And then this 2b here, we're going to subtract b+b. So let's go ahead and stack all those vectors together. We know that a is going to be 1 to the right and one up. So that means 3a is just going to be if I stack 2 more on top of those. So this whole entire thing here ends up being 3a. And so, now we just have to add the negative 2b. Well, my regular b vector is going to be 3 to the right and one up. So from the tail of this one, from the end of this one, I have to go not 3 to 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 going to be my negative b vector. So this is my negative b. And I have to do it again. I have to do another 3 to the left and then one down. So I'm going to go this way. So notice how we've got this b 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 resultant vector here is going to be from the start of the first to the end of the last one. It's basically as if I just walked in this path. And so you're going to just draw the shortest path between those. This is my c vector and then you just count up the legs. This is 3 and this is 1 and so the magnitude which is given by the absolute value sign is 32 + 12 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

Here’s what students ask on this topic:

How do you add vectors graphically using the tip-to-tail method?

To add vectors graphically using the tip-to-tail method, follow these steps: First, draw the first vector as an arrow. Then, place the tail of the second vector at the tip of the first vector. Continue this process for any additional vectors. The resultant vector is the arrow drawn from the tail of the first vector to the tip of the last vector. This resultant vector represents the shortest path from the start to the end point. The magnitude of the resultant can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

What is the resultant vector in vector addition?

The resultant vector in vector addition is the vector that represents the shortest path from the start of the first vector to the end of the last vector when vectors are added using the tip-to-tail method. It effectively combines the magnitudes and directions of all the vectors being added. The magnitude of the resultant vector can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

How do you subtract vectors graphically?

To subtract vectors graphically, you use the tip-to-tail method but reverse the direction of the vector being subtracted. For example, if you want to subtract vector B from vector A (A - B), you first draw vector A. Then, instead of adding vector B, you add the negative of vector B, which points in the opposite direction. The resultant vector is the shortest path from the start of vector A to the end of the negative vector B. The magnitude can be calculated using the Pythagorean theorem: $\sqrt{a^2+b^2}$.

What happens when you multiply a vector by a scalar?

When you multiply a vector by a scalar, the magnitude of the vector changes, but its direction remains the same. If the scalar is greater than 1, the vector's magnitude increases, making it longer. If the scalar is between 0 and 1, the vector's magnitude decreases, making it shorter. If the scalar is negative, the vector's direction reverses. For example, multiplying vector A by 2 (2A) doubles its length, while multiplying it by 0.5 (0.5A) halves its length.

Does the order of adding vectors affect the resultant vector?

No, the order of adding vectors does not affect the resultant vector. This property is known as the commutative property of vector addition. Whether you add vector A to vector B (A + B) or vector B to vector A (B + A), the resultant vector will be the same. The magnitude and direction of the resultant vector remain unchanged regardless of the order in which the vectors are added.