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 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,
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
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