Main content

## Precalculus

### Unit 6: Lesson 5

Vector addition and subtraction# Adding & subtracting vectors end-to-end

Build intuition behind adding and subtracting vectors visually and the "end-to-end" method.

## Want to join the conversation?

- At last in the video Sal said that vector a + vector b= -vector c , then it can be also written as vector a + vector b + vector c = 0 , but here will it be scalar 0 or vector 0 ?(31 votes)
- Vector addition is closed, meaning vector addition results in a vector. The answer to your question is the zero vector.(38 votes)

- If I am given a triangle whose 2 sides are denoted by vector a and vector b, what would the third side?

Should it be( a+b) Or( a-b) ?(3 votes)- It depends on how the vectors are oriented.

If, for example, 𝒂 and 𝒃 originate from the same point, and 𝒄 goes from the endpoint of 𝒃 to the endpoint of 𝒂, then 𝒄 = 𝒂 − 𝒃.

But, if 𝒃 starts at the endpoint of 𝒂, and 𝒄 goes from the starting point of 𝒂 to the endpoint of 𝒃, then 𝒄 = 𝒂 + 𝒃.

In total there are 8 different constellations, and depending on which one it is 𝒄 can be either 𝒂 + 𝒃, 𝒂 − 𝒃, 𝒃 − 𝒂, or −𝒂 − 𝒃(14 votes)

- what if vector a and b are perpendicular? would we use pythagoras' theorem or vector addition?(7 votes)
- If we are trying to find the distance from the starting point to the end point, then we would use the Pythagorean theorem. In other words, if we are trying to find the magnitude of the vector, we would use the Pythagorean theorem.

If we are trying to find the displacement, then we would use vector addition.(1 vote)

- Can we draw a scalar? My math teacher says that we can't draw a scalar, but I think that the side of a triangle, for example is a scalar.(3 votes)
- Basing my answer off of what my physics teacher had said, distance --the shortest path from one point to another -- is a scalar. You can draw this path with a straight line segment. So, I would argue that you can draw a scalar as a line segment.(3 votes)

- does that mean that the magnitude of a vector is the sum of two vectors?(2 votes)
- No, a vector can be said to be the sum of one or more component vectors, but the magnitude of the vector is equal to sqrt((x^2) + (y^2)) where x and y are the component vectors. (This can also be extended to calculate the magnitude of vectors in more than 2 dimensions.)(3 votes)

- Doesn't make any sense to me. A vector is magnitude + direction, I can understand adding magnitudes but what does it mean to add direction? I cannot wrap my head around it. And why is the length of vector C not equal to length of vector A + length of vector B?(2 votes)
- UPDATE: I think I get it now. Sal gives a better explanation of what's happening in the "Unit vectors intro" video.(2 votes)

- 4:54i think he showed wrong direction of resultant vector c it should be from tail of a to head of b which is shown wrong i think(0 votes)
- You might want to listen more closely to what Sal was saying. In the second example he was working, the vector C he was using was NOT the sum of vectors A and B. Once he switched directions as you suggest to create the vector -C he then set up the equation A + B = -C.(8 votes)

- Why do we take -c instead of +c?(2 votes)
- It has to do with c's starting point. In the first problem, A+B=C, The terminal end of A,(the end with the arrow) is the initial end of B. C shared a starting point with A and a terminal point with B.

Now look at how he drew it for A -B = D, and you should start to get it. I recommend that you see if you can figure out why the direction matters. Its good exercise for the brain.(2 votes)

- At4:51, Sal says it's going in a circle. Isn't it going in a triangle?(1 vote)
- Yes, I think he is just saying "circle" to show that it's going "around" and you will eventually end up where you started, able to repeat the cycle again.(3 votes)

- So are vectors not side lengths as a+b=c which isn't always possible?(2 votes)

## Video transcript

- [Voiceover] Let's build our
intuition for visually adding and subtracting vectors. So let's say that I have
vector a, and I add that to vector, to vector b, and the resulting vector is a vector c, is vector c. So what could this look like visually if we assume that a, b and c
are two-dimensional vectors? Well, I'm just going to draw
what vector a might look like. So let's say that this
right over here is vector a, and vector b, vector b, since I'm adding it to
vector a, I'm going to put its initial point at the
terminal point of vector a, and then I'm going to draw vector b. So let's say vector b looks like that. So that is vector b, let me label these. That is vector a, this is vector b. And I did that so that I can
figure out what the sum is, what vector c is going to be. So that is vector b,
and what would that be? Well, we would start at the
initial point of vector a and then go to the
terminal point of vector b. So this right over there would be the sum. So that would be vector
c right over there. So the important realization
is if I add two vectors, I would put the tail of one
at the head of the other. And now what's neat
about this if I'm adding, the order doesn't matter. I could've done this the other way around. I could've started with vector b. I could've said vector b plus vector a is equal to vector c,
is equal to vector c, and you could see that visually. It would be a slightly
different visual diagram but you get to the same place. So if I start with vector b,
let's say I start over here, In fact, you don't have
to start at the origin but let's say that was the origin. So I could start with
vector b, draw vector b just like that, and
then add vector a to it. So start vector a at the
terminal point of vector b, and then go to (mumbles)
just draw vector a. So vector a. So once again, a vector,
I can shift them around as long as I'm not changing the direction or their magnitude. So vector a looks like that. And notice, if you now go and
start at the initial point of b and go to the terminal
point of a, you still get vector c. So that's why a plus b and b
plus a are going to give you the same thing. Now what if instead of saying a plus b I wanted to think about what
a minus b is going to be? So let me write that down. Vector a minus vector b, minus vector b. And let's call that vector d. That is equal to vector d. So once again, I could start with vector a and here, order matters. So vector a looks something like this. this is hand-drawn so it's not going to be completely perfect. So vector a, just like that. And one way of thinking
about subtracting vector b is instead of adding vector
b the way we did here, we could add negative b. So negative b would
have the same magnitude but just the opposite direction. So that's vector a. Vector negative b will
still start right over here, but will go in the opposite direction. So let's do that. So negative b is going to look like this, is going to look something like this. So that is negative b. Notice, same magnitude
exactly opposite direction. We've flipped it around 180 degrees, and now the resulting
vector is going to be d. So vector d is going to look like that, vector d. So c is a plus b, d is a minus b. Or you can even call this
a plus, a plus negative b. Now with that out of the
way, let's draw some diagrams and go the other way. See if we could go from the diagrams to the actual equations. So let's start with, let
me draw an interesting one. So let's say that this... Let's say that's vector a. Vector a. I'll use green. Let's say that, that is vector b, and I will now use magenta. And let's say that this is vector c. Vector c. So I encourage you to pause the video and see if you can write an equation that defines this relationship. Well, this is interesting
because they're all going in a circle right over here. Let's say that you started at... This is your initial point. You said, okay, a plus b, well, if you're trying to
figure out what a plus b is going to be, the resulting
vector would start here and end here. But vector c is going in
the opposite direction. But we could, instead of thinking
about vector c like this, we could think about the
opposite of vector c, which would do, so instead of calling this
c, I could flip this around by calling it this negative c. So I could flip this around, and now, let me do the same color, this would be equal to negative c. Notice, before I just had vector c here and it started at this point
and ended at this point. Now I just flipped it
around, it has the exact opposite direction, same magnitude. Now it is negative c. And this makes it easier for
us to construct an equation because negative c starts at
the tip, at the initial point or the tail of vector a,
and it goes to the head of vector b or the
terminal point of vector b. So we can now write an equation. We could say vector a plus vector b, plus vector b, is equal to, is equal to not c, it's equal to the negative of vector c. So hopefully, you found that interesting.