Main content

## Precalculus

# Finding the components of a vector

Sal finds the x and y-components of a vector given its graph.

## Want to join the conversation?

- Why did he start from A (4,4) instead of B (-7,-8)(21 votes)
- Remember, in a vector, there is a specific beginning and ending point, and the ending point is marked as an arrow. The reason an arrow is used is because a vector uses magnitude, the amount something moves, or the speed with which it moves, and direction. In this case, the direction is left and down. Does that help? I hope it did.(64 votes)

- When it comes to solving vector components, I am still confused when to add or subtract the points.(12 votes)
- You ALWAYS subtract the points. The thing is you subtract ENDING POINT - STARTING POINT. The problem you're given will define the direction of the vector. So, if the direction defined by the problem is "A to B", you subtract Point B - Point A. If the direction is defined by the problem as "B to A", you subtract Point A - Point B.(6 votes)

- I've seen any questions of this kind that don't provide graphs like this. So if we are to find components like that of the first question in the video without the graph, how do you do it?(3 votes)
- All you really need are the points. From there you find your change in x and change in y through subtraction or construction of your own graph.(6 votes)

- What's the point of knowing the components? Does it help in determining if two points have the same direction?(4 votes)
- Yep, you use the components and trigonometry to determine the angle of the vector, and that tells you the direction(1 vote)

- How do you know what direction to move the arrows, right or left? It depends on what?(2 votes)
- There are two hints here as to the direction of the vector AB:

1) The arrow over the top of AB indicates that the vector starts at A and terminates at B; and 2) If you look at the diagram, it'll show that the arrow does, in fact, "point" toward the point B.(3 votes)

- but why did we go in that specific direction? can we not go in the other direction and get different values for the components? and does this mean that change in x and change in y are the components?(2 votes)
- You could go down from point A and go left to reach point B (if that's what you meant by "other direction"). But, observe that the vectors you get will still be the same and hence, a vector cannot have two sets of components if taken in two different directions.

Change in x and change in y are indeed the components. They cumulatively make up the change in position from A to B, which is*exactly*what the vector is showing us here.(2 votes)

- So I got this question asking: Vector AB has a terminal point (4,-7), an x component of -3 and a y component of -9. FIND THE COORDINATES OF IT'S INITIAL POINT. Is there a video on questions like this.(1 vote)
- A vector is equal to its terminal point minus its initial point. Therefore, we have

(-3, -9) = (4, -7) - initial point

initial point = (4, -7) - (-3, -9) = (4 - (-3), -7 - (-9)) = (7, 2).

The initial point is (7, 2).

Have a blessed, wonderful day!(4 votes)

- Around the end of the video, he concludes that the change in x would be -11, and the change in y would be -12. However, a Change in the values wouldn't be described with a negative value. Wouldn't the change in x and y be ( 11, 12 ) ?

Thanks for any help in advance.(2 votes)- In this example we are working on the coordinate plane. To move to the left on the x-axis and down on the y-axis would be to move down in numbers, thus negative.(1 vote)

- why we call the the change in x and the change in y component of the vector ?

is it because we could constrict the vector by knowing them ?(2 votes) - In my Trigonometry class, my teacher specifically said to use Chevron brackets, which are ⟨ ⟩, when writing component form. Shouldn't the vector be ⟨-11,-12⟩ instead of (-11,-12)?(1 vote)
- That is an effective way to show that you are talking about a vector and not an ordered pair, but its more a matter of choice than a mathematic rule. Plus, every point on a graph can be described as the terminal end of a position vector, so an ordered pair can be thought of as the end of a vector.

I wouldn't worry too much about it. Just follow the rules in your classes, but remember that some of them aren't universal.(3 votes)

## Video transcript

- [Voiceover] Find the
components of vector AB. So when they're talking
about the components, at least in this context, they're just talking about
breaking it down into if we start at point A and
we're finishing at point B, how much do we have to
move in the X direction? So this is going to be
essentially our change in X. And then how much do we have
to move in the Y direction to go from point A to point B? So this one over here is
going to be our change in X. I just wrote the Greek
letter Delta for change in X. And then, the second
component is going to be our change in Y. And to think about that, let's just think about what our starting and final points are, our initial and our terminal point are. So, this point right over here, point A, its coordinates are (4,4). And then point B, its coordinates are, let's see its X coordinate is (-7,-8). So let's first think about
what our change in X is, and like always, I encourage
you to pause the video and try to work through it on your own. Well let's see, if we're starting at four and then we are going from X equals four. That's where we're starting, to X equals negative seven. So that right over there
is our change in X. And there's a couple of
ways you could compute that. You could say, "Look, we
finished at negative seven. "We started at negative four." You take your final point
or where you end up, so that's negative seven, and you subtract your
initial point, minus four, which is going to be equal to negative 11. The negative tells us that
we decreased in X by 11. And you could see that. If you could just visually
count the squares, you could say, "Look,
if I'm going from four "to negative seven, I have to go down four "just to get back to X equals zero, "and then I have to go down another seven. "So I have to go to the left 11 spaces." So that's negative 11. So that's my X component, negative 11. And what is my change in Y? Well I'm going from Y equals four. In fact, I'll start at
this point right over here. I'm starting at Y equals four. And I'm ending up at Y is equal to, let me do that in that other color. So, I'm starting at Y is equal to four, and I'm ending up at Y is
equal to negative eight. So our change in Y, our change in Y, what's
going to be my final Y value, which is negative eight, minus my initial Y value, which is four, minus four, which is equal to negative 12. So negative 12. And you could see that here. If I'm starting up here, I have to go four down just
to get back to the X axis. Then I have to go down another eight, so I have to go down a total of 12. And you can see something interesting that I've just set up here. You could also view this bigger vector. Vector AB is being constructed of this X, this vector that goes
purely in the X direction, and this vector that goes
purely in the Y direction. If you were to add this red
vector to this blue-green, dark blue-green vector, you would get vector AB, but we'll talk more about
that in future videos.