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### Course: Precalculus > Unit 6

Lesson 5: Vector addition and subtraction# Adding & subtracting vectors

Sal shows how to add vectors by adding their components, then explains the intuition behind adding vectors using a graph.

## Want to join the conversation?

- Why do vectors "combine" to form new vectors like that?(15 votes)
- The intuition behind this "combination" is that the resultant vector of ,say, 2 vectors would be the addition of those vectors.

Example : If the displacement of a person is 5 miles east ,and then 2 miles south ,their resultant displacement vector would be the sum of the 2 previous vectors.(51 votes)

- Don't we put the X and Y values in a matrix sort of form? Where the X is above and the Y below?(10 votes)
- I think you are referring to the vector multiplication. Here Sal is talking about addition and subtraction.(4 votes)

- I am stuck on the following:

eg. u (-6, -6) w (-8, -7)

u-w = (-6 (-8), -6 (-7))

I dont know to graphically represent this! any help?(5 votes)- Instead of thinking it as subtracting w think of it as adding negative w. So negative w is like scaling w by -1 which you probably learnt in one of the previous videos. This makes (-8*-1,-7*-1)=(8,7). So take the vector u and add the vector -w to u. the way to graph it is just graph u from the origin and then graph -w by placing the initial point at the terminal point of u and drawing a line from the initial point of u which is the origin to the terminal point of -w which would be at (2,1). So when subtracting the two vectors, the new vector is equal to a x component of 2 and a y component of 1.(3 votes)

- Hey when do add and subtract vectors? like in what scenarios would you add the vectors or subtract them?(5 votes)
- You would add and subtract vectors if you were trying to plot the direct route to a certain point. Say, Bob went north 9 meters and then went East for 12 meters. 9m @ 90° + 12m @ 0° = 15m @ 36.87°

So you could go 15m at a 36.87° angle to get to Bob "as the Crow flies."(4 votes)

- When adding vectors, do you have to always write it out like how he did it in the video, or could you just be a little quicker and do it in your head? Is there a difference?(3 votes)
- if you are just doing a calculation that you are comfortable with, it makes little sense to write it out, when you can just do it in your head. But if you are not completely adept at the skill, you will make mistakes occasionally, and writing it down lets you check your process to see where/how/why the mistake was made.

You dont learn from doing calculations in your head, you learn from making mistakes, and then figuring out why.(8 votes)

- and why do we also use matrices for vectors operations ?(2 votes)
- Vectors can be seen as nx1 matrices, So vector operations are basically an extension of matrix operations(4 votes)

- I don't understand what Sal means when he says that these vectors are 2 Dimensional (I'm pretty sure he mentioned this in another video). What does 2D mean in this circumstance? What would 3D look like?

Can someone please help.(2 votes)- Another way to think about it is the number of ways that a shape can move. A 3D shape can move up/down, left/right, and backwards/forwards (3 ways). A 2D shape can move up/down and left/right (2 ways). A 1D shape can move left/right (1 way). And a 0D shape cannot move at all. The only thing to remember with this way is that a dimension is not restricted to one direction of movement, so a shape that can move in/out (however that works) is in the 1D just as much as a shape that can move left/right.(3 votes)

- At0:05, Sal says
*'2-dimensional vectors'*.

Do**3-dimensional vectors**exist?(3 votes)- Yes. There are vectors in any number of dimensions, including infinite-dimensional vectors.(2 votes)

- Why is it that you can add vectors this way? In triangles, two sides added together must be greater than the remaining side. I understand that you can prove it using a parallelogram method but I am still very confused. Can someone explain this in layman's terms?(3 votes)
- Adding vectors involves combining the direction and magnitude of two vectors to find a resultant vector. This is done using the parallelogram method, which involves drawing the two vectors as sides of a parallelogram and then finding the diagonal that connects the two endpoints. The resultant vector is equal in magnitude and direction to this diagonal.

The reason why this method works is due to the properties of vector addition. Unlike the sides of a triangle, which have a fixed length and cannot be changed, vectors can be translated and shifted in space without affecting their properties. When we add two vectors, we are effectively translating and shifting them so that their endpoints meet, and then finding the vector that connects the initial point of the first vector to the final point of the second vector.

This resultant vector represents the combined effect of both vectors and is equivalent to the diagonal of the parallelogram formed by the two vectors. The parallelogram method works because the opposite sides of a parallelogram are parallel and equal in length, meaning that the magnitude and direction of the diagonal can be easily determined using basic geometry.

In summary, the reason why we can add vectors using the parallelogram method is due to the properties of vector addition and the fact that vectors can be translated and shifted in space without affecting their properties. By drawing the two vectors as sides of a parallelogram, we can find the resultant vector by finding the diagonal that connects the two endpoints.(2 votes)

- At5:03why he added two and three to shift i mean i didn’t understand how ?(2 votes)
- If we assume that the terminal point of vector a as origin then it makes sense . Sal moved 2 steps ( 2 units ) to the right from terminal point of vector a because the X coordinate of vector b is 2 , then he moved 3 steps ( 3 units ) up from the terminal point of vector a because the y-coordinate of vector b is 3. A vector can be placed any where in the plane without changing its magnitude and direction . In order to understand better we assume that the initial point of a vector as origin ( to make things simple )

Does it make sense ?(4 votes)

## Video transcript

- [Voiceover] So let's get
some practice and hopefully a little bit of intuition
for adding and subtracting two dimensional vectors. So let's say that I have vector A, and let's say that its x component is I don't know, three, and it's
y component is negative one. And let's say we also have a vector B. Vector B, and let's say
that its x component is two, and let's say
its y component is three. Now, let's first think about what would A plus B equal? So what resulting vector would this be? A plus B is equal to what? And I encourage you to pause the video and think about what this would be. Well, the convention
is, is if we're taking the sum of two vectors, we can just add up their x, their x components to get our new x component and up their y components to get the new y component. So the x component of
vector A plus vector B is going to be three plus two. That's going to be the x
component, which we know is five. And the y component is
going to be negative one. Negative one plus three. Negative one plus three,
and so the resulting vector is going to have an x component of five and a y component
of negative one plus three is equal to two. Alright, that was pretty straightforward. Now instead of adding
them, let's think about what would happen if we subtract them. So let's think about what A, what vector A minus vector B would be,
and I suspect that you might be able to guess or at least think about what would happen if you subtract vector B instead of adding vector B. Well, as you might have
guessed, instead of adding the corresponding
components, we subtract it. So our x component is going to be the x component of A minus
the x component of B 'cause it's vector A minus vector B. So our x component is going to be three minus two. Three minus two, and our y component is going to be the y component of A, negative one, minus the y component of B. So minus three. Minus three, and so the resulting vector is going to be, is going to
be the vector three minus two. The x component's going to be one, and then the y component of negative one minus three is negative four. Now what I just showed
you, this is the convention for adding and subtracting
two dimensional vectors like vectors A and B. Let's think a little bit about how we can visually depict what is going on. So let's first visually depict
adding vector A and vector B. So let me draw some axes, and so my, so this could be, there's my y-axis. Y-axis, let's see. The highest y value I get to is, the highest y value I get to is three, and the lowest is negative four. And then, so let me draw
the x-axis some place. Whoops, didn't mean to do that. I did the zoom by accident. Alright. So the x-axis, put it right over there. There's my x-axis, and let's see. The highest x value is five. The lowest is three. Actually, I could just focus on the, I could just focus on
the, let me just focus on the right hand side. So that's y, and then let me do x looking like that. X-axis. And now let's see. Vector A is 3, -1. So one, two, three, and negative one. If we want to just visualize
it in standard form, we could put its initial
point at the origin and its terminal point at the point 3, -1. And so we could draw it like that. We could also shift it
around, as long as it has the same, as long as it
has the same magnitude and direction, we could shift it around. So that right there is our vector A. Now Vector B, 2,3, we could
have its initial point at the origin and just, we
could draw it like this. So the x-coordinate is two. Y-coordinate one, two, three. So its terminal point could be there. So we could just draw it
like this for vector B, but if we're going to
add vector B to vector A like we have right over
there, what we want to do is shift vector B over so that
its, I guess you could say, tail starts at A's head,
or its initial point starts at A's terminal point. So let's do that. So if we start right over
here, we're going to go two in the x direction. So we're going to go two
more in the x direction, and we're going to go
three in the y direction. So one, two, three. So we're going to end up right over there. So notice, this is the
same vector, vector B. I've just shifted it over. Has the same magnitude and same direction as what I had drawn before, as that vector right over there. I have just shifted it
down into the right. And now we do this, so this is vector A, I'm adding Vector B. I put the, I put the
initial point of vector B at the terminal point of vector A. I've shifted it over right over there. So now I can figure out
the resulting vector, A plus B, the resulting vector A plus B, by going from the initial point of A to the terminal point of B. So it's gonna start here,
and then go over there, and notice, this vector I'm
going five in the x direction, and two in the y direction. This is the vector 5,2 right over there. That is vector A plus vector B. But also think about what
happens when we do A minus B. Well, we could still have
vector A just like we drew it, but now instead of
putting vector B's tail or initial point at the
terminal point of vector A, we would wanna put negative B. So negative B would just
be, have the same magnitude but it'll be in the opposite direction. So instead of going two
to the right and three up, we would go two to the
left and three down. So just two the left and
one, two, three down, and we would end up right about there. And so, if you subtract A minus B, that's the same thing
as A plus negative B. So negative B is going to
look something like this. It's going to be the
exact opposite direction. It's gonna look like that, and so there if you start at the
initial point of A and get to the terminal point of the
negative B now, this is all hand-drawn, so it could be
a little bit more precise, notice we get to the point 1, -4. 1, -4. So this vector right
over here, let me do that in a different color, this brown, whoops, that's not what I wanted to do. This brown vector right
over here, that is the vector A minus B, and then this white one, actually, let me do this
in another color as well, this magenta vector right
here, that is A plus B. So hopefully that makes sense, and if you, given the
components, you're just, if you're adding the vectors, add
the correspondent components. If you're subtracting the
vectors, well if you're subtracting B from A,
subtract B's components, corresponding components from the corresponding components of A.