All the problems we've been dealing with so far have essentially

been happening in one dimension. You could go forward or backwards.

or right or left, or you could go up or down.

What I want to talk about in this video is what happens when we have two dimensions

or three or four, or really an arbitrary number of dimensions

although if we're dealing with mechanics

we normally don't have to go more than three dimensions.

But we're going to deal with more than one dimension, or two dimensions,

we're also going to be dealing with two-dimensional vectors.

I just want to make sure that through this video

that we understand at least the basics of two-dimensional vectors.

Remember, a vector is something that has both magnitude and direction.

The first thing I want to do is give you a visual understanding

of how vectors in two dimensions would add.

So let's say that I have a vector here that is Vector A,

and once again, its magnitude is specified by the length of this arrow

Its direction by the direction of the arrow, so it's going in that direction.

I have another vector, called Vector B

It looks like this,

What I want to do in this video is to think about what happens when I add

Vector A to Vector B. So there's a couple of things to think about when you visually depict vectors

The important thing is for example, Vector A, that you get the length right,

and also get the direction right. Where you draw it doesn't matter,

So this could be Vector A, and this could also be Vector A.

Notice, it has the same length, and also the same direction.

I could draw it up here, or there; it does not matter.

I could draw Vector B over here

It's still Vector B; it still has the same magnitude,

and the same direction.

We're not saying its tail has to start at the same place that Vector A's tail starts.

I could draw Vector B over here

So I can always have the same vector, but I can shift it around.

As long as it has the same magnitude, the same length, and the same direction.

And the whole reason I'm doing that is because the way to visually add vectors,

If I wanted to add Vector A,

plus Vector B,

and I'll show you how to do it more analytically,

in a future video

I can literally draw Vector A,

And then I can draw Vector B, but I put the tail of Vector B to the head of Vector A.

I shift it so it's tail is right at the head of Vector A.

so it will look something like this.

And then if you go from the tail of A,

all the way to the head of B,

and you call that Vector C,

That is the sum of A and B.

And it should make sense.

If you think about it, let's say these were displacement vectors,

so A shows you're being displaced this much in this direction

B shows you're being displaced this much in this direction, the length of B in this direction

And if I were to say you have a displacement of A

And then you have a displacement of B,

what is your total displacement?

So you would have to be shifted this far in this direction,

and then shifted this far in this direction

so the net amount that you've been shifting is this far in this direction

This would the be the sum of those.

Now we can use that same idea to break down a vector in two dimensions,

and I'll give you a sense of what that means in a second,

If I have Vector A, let's call this Vector X

I could say that Vector X is going to be the sum of

This vector right here in green,

and this vector in red

Notice, if X starts at the tail of the green vector,

and goes all the way to the head of the magenta vector,

and if the magenta vector starts at the head of the green vector,

and then finishes at where Vector X finishes,

Hopefully from this explanation right here, you can see

green vector plus magenta vector gives us this X vector

I put the head of the green vector to the tail of this magenta vector

but the whole reason why I did this is that

if I can express X as the sum of these two vectors

it then breaks down X into it's vertical component

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