What I want to do in this

video is show you a way to

represent a vector by its component

And this is sometimes called engineering notation for vectors.

But its super useful because

it allows us to keep track

of the components of the vector, and

it makes it a little bit tangible when we

talk about the individual components.

So lets break down this vector right over here.

I'm just assuming it is a velocity vector, vector v ,

its magnitude is 10 m/s and its pointed in the direction

30 degrees above, above the horizontal.

So we have broken down these

vectors in the past before.

The vertical component right

over here, its magnitude,

its magnitude would be, so, the magnitude of

the vertical component right over here,

is going to be 10 sin of 30 degrees,

is going to be 10 meters per sec times

the sin of 30 degrees

, sin of 30 degrees, this just comes

from basic trigonometry from soh cah toh,

and I covered that in more details

in previous videos

sin of 30 degrees is 1/2

So this is going to be 5 or 5 meters per second

Ten times 1/2 is 5, 5 meters per second so

that is the magnitude of its vertical component

And in the last few videos I kind of

, in a less tangible way of specifying

the vertical vector, I often used this notation

which isn't that tangible as I like it,

that's why I am going to make it little bit

better in this video.

I said that the vector

its self is 5 meters per sec, 5 meters per sec

but what I told you that the direction is

implicitly given because this

is a vertical ,

this is a vertical vector and I told you

in previous videos that

if its positive, it means up

and if its negative its means down.

So I kind of have to give you this context

here so that you could appreciate

that this is a vector, that just the sign

of it is giving you its direction

But I have to keep telling you this

a vertical vector.So its a little bit

it wasn't that tangible,and so we had the same issue,

when we talked about the

we had the same issue talked about

the horizontal vectors, so this horizontal vector

right over here, the magnittude of it,

the magnitude of this horizontal vector is going

to be 10 cosine of 30 degres.

And once again comes straight out of basic trigonometry.

tan cosine of 30 degrees and so

cosine of 30 degres is

square-root of three over 2

square root of 3 over 2.

multiply it by ten, you get

5 square roots of 3 meters per sec.

And once again in previous videos

I said, look this is actually

I used this notation sometimes

where I was actually saying the vector is

5 square root of 3 metres per sec

but in order to ensure that this wasn't not just the

magnitude I kept having to tell you that

in the horizontal direction if its positive

, its going to the right and

if its negative its going to the left.

But what I want to do in this video

is give us a convention so that I don’t have to

keep doing this for the direction

and it all, it makes all a little bit more tangible

And so what we do is we introduce

the ideas of, or the idea of unit vectors.

Of unit vectors.

so by definition we introduce the

vector i, the vector i, sometimes its called

i hat, and I'll draw it like here.

So the vector.Let me make it a little bit smaller,

So the vector i hat,

so that right there is a picture of the vector i hat

we put a hat on top of i

to show that it’s a unit vector.

And what a unit vector is,

so the i hat vector goes in the

positive x -direction.

That‘s just how its defined

and we also, unit vector tells us

that its magnitude is one.

So, the magnitude of the vector i hat

is equal to one and its direction

is in the positive x -direction.

So if we really wanted to specify

this kind of x -component vector in a better way.

We really should call it

, we really should call it,

five square roots of 3 times this unit vector.

Because it 5, this green vector over here

is going to be 5 squared roots of 3

times this vector right over here.

cause' this vector just has length 1.

So its 5 squared of 3 times the unit vector.

and what I like about this is that

now I don’t have to, tell you

Remember this a horizontal vector,

positive is,

positive is to the right and

negative to the left,

It’s implicit here,

because clearly if it’s a positive value

Its going to be a positive multiple of i,

its going to go to the right

If its a negative value

it flips around the vector and

its goes to the left .

So this is a actually a better way of specifying,

of specifying,

the x component vector

or if I broke it down this vector v,

into its x components

this is a better way of specifying that vector.

Same thing for the y -direction,

We can define a unit vector

and let me pick a color,

that I have not used yet,

let me find a, oh, this pink I haven’t used.

We can define a unit vector

that goes straight up in the

y-direction called unit vector j

and once again the magnitude of unit vector j

is equal to 1

This little hat on top of it tells us

or sometimes is called a caret,

a caret character,

tells us that it is a vector but

it is a unit vector

and has magnitude of 1.

And by definition the vector j

goes in, has a magnitude of 1

in positive y-direction, so this

the y -component of this vector,

instead of saying its,

5 meters per second in upwards direction

and instead of saying that its implicitly upwards

because the vertical vector or its

vertical component in its positive,we can now be a little bit more

Or a little bit more specific about it ,we could

say it’s a equal to

equal to 5 times j

, 5 times j

because you see this magenta vector,

is going the exact same direction as j

, the exact same direction as j

it is just 5 times longer,

I don’t know if its exactly 5 times,

I'm trying to estimate it right now.

Its 5 times longer

Now what's really cool about this, is besides

just being able to express the components as

now multiple of explicit vectors,

instead of just being able to do that

which we did do, or we are

representing the components as explicit vectors

we also know that the vector, v

is the sum of its components,

if you add, if you start with this, this

green vector right here

and you add this vertical component

right over here you have head to tails

you get, you get the blue vector,

and so we can actually use the components

to represent the vector itself

we don't always have to draw like this

So we can write,

that vector, v is equal to

its equal to vector,

let me write it this way, is equal to its x-component

vector plus the y- component vector

, plus the y- component vector,

And we can write that, x-component vector

is 5 square roots of 3 times i

, 5 square roots of 3 times i,

and then its going to be plus

the y component, the vertical component

which s five times j,

which s five times j

and so what's really neat here

is now you can specify any vector

in two dimensions,

by some combintion of i and j

scaled up combination of i and j

and if you want to go in three dimension, and you

often will,

as specially physics class moves on through the year

you can introduce a vector in the positive z-direction

depending upon how you want to do it,

although z is normally up and down,

but whatever the next dimension is

you can divine, divide a vector k

that goes into that third dimension

here I will do it in a kind of unconventional way

I'll make k go in that direction.

Although the standard convention when you do

in the three dimensions is that k is the

up and down dimension.

But this by itself is already petty neat because

we can now represnt any vector,

any vector through its components

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