Unit vectors and engineering notation Video Lecture - Class 11

What I want to do in
this video is show you
a way to represent a
vector by its components.
And this is sometimes
called engineering notation
for vectors.
But it's 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 let's break down this
vector right over here.
I'm just assuming it's
a velocity vector.
Vector v. Its magnitude
is 10 meters per second.
And it's pointed in a
direction 30 degrees
above the horizontal.
So we've broken down these
vectors in the past before.
The vertical component
right over here.
Its magnitude would
be-- so the magnitude
of the vertical component,
right over here,
is going to be 10
sine of 30 degrees.
It's going to be 10 meters per
second times the sine of 30
degrees.
This comes from the basic
trigonometry from sohcahtoa.
And I covered that in more
detail in previous videos.
Sine of 30 degrees is 1/2.
So this is going to be 5,
or 5 meters per second.
10 times 1/2 is 5
meters per second.
So that's 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 use this notation, which
isn't as tangible as I like.
And that's why I'm
going to make it
a little bit better
in this video.
I said that that vector
itself is 5 meters per second.
But what I told you is that
the direction is implicitly
given because this
is a vertical vector.
And I told you in previous
videos that if it's positive,
it means up and if it's
negative, it means down.
So I kind of have to give you
this context here so that you
can appreciate that this is a
vector that just the sine of it
is giving you its direction.
But I have to keep telling
you this is a vertical vector.
So it wasn't that tangible.
And so we had the
same issue when
we talked about the
horizontal vectors.
So this horizontal
vector right over here,
the magnitude of this
horizontal vector
is going to be 10
cosine of 30 degrees.
And once again, it
comes straight out
of basic trigonometry.
10 cosine of 30 degrees.
And so cosine of 30 degrees is
the square root of 3 over 2.
Multiply it by 10,
you get 5 square roots
of 3 meters per second.
And once again, in
previous videos,
I used this notation
sometimes, where I was actually
saying that the vector
is 5 square roots of 3
meters per second.
But in order to ensure that
this wasn't just a magnitude,
I kept having to tell you
in the horizontal direction.
If it's positive it's
going to the right,
and if it's negative,
it's going to the left.
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 makes it all a
little bit more tangible.
And so what we do
is we introduced
the idea of unit vectors.
So by definition, we'll
introduce the vector i i.
Sometimes it's called i hat.
And I'll draw it like here.
I'll make it a
little bit smaller.
So the vector i hat.
So that right there is a
picture of the vector i hat.
And we've put a little
hat on top of the i
to show that it
is a unit vector.
And what a unit
vector is-- so i hat
goes in the positive
x direction.
That's just how it's defined.
And the unit vector tells
us that its magnitude is 1.
So the magnitude of the
vector i hat is equal to 1.
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
5 square roots of 3 times
this unit vector.
Because this green
vector over here
is going to be 5
square roots of 3 times
this vector right over here,
because this vector just
has length 1.
So it's 5 square roots of
3 times the unit vector.
And what I like about
this is that now
I don't have to
tell you, remember,
this is a horizontal vector.
Positive is to the right,
negative is to the left.
It's implicit here.
Because clearly if this
is a positive value,
it's going to be a
positive multiple of i.
It's going to go to the right.
If it's a negative value,
it flips around the vector
and then it goes to the left.
So this is actually a better way
of specifying the x component
vector.
Or if I broke it down, this
vector v into its x component,
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 this
pink I haven't used.
We can find 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 it's called a
caret character-- that tells us
that it is a vector,
but it is a unit vector.
It has a magnitude of 1.
And by definition, the vector
j goes and has a magnitude of 1
in the positive y direction.
So the y component
of this vector,
instead of saying it's
5 meters per second
in the upwards direction or
instead of saying that it's
implicitly upwards because
it's a vertical vector
or it's a vertical
component and it's positive,
we can now be a little bit
more specific about it.
We can say that it is
equal to 5 times j.
Because you see, this
magenta vector, it's
going the exact same direction
as j, it's just 5 times longer.
I don't know if it's
exactly 5 times.
I'm just trying to
estimate it right now.
It's just 5 times longer.
Now what's really
cool about this
is besides just being able to
express the components as now
multiples of explicit
vectors, instead of just being
able to do that--
which we did do,
we're representing
the components
as explicit vectors--
we also know
that the vector v is the
sum of its components.
If you start with this
green vector right here
and you add this vertical
component right over here,
you have head to tails.
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 it like this.
So we can write that
vector v is equal to-- let
me write it this way-- it's
equal to its x component
vector plus the y
component vector.
And we can write
that, the x component
vector is 5 square
roots of 3 times i.
And then it's going to
be plus the y component,
the vertical component, which
is 5j, which is 5 times j.
And so what's really neat here
is now you could specify any
vector in two dimensions
by some combination
of i's and j's or some scaled
up combinations of i's and j's.
And if you want to go
into three dimensions,
and you often will, especially
as the physics class
moves on through
the year, you can
introduce a vector in
the positive z direction,
depending on how
you want to do it.
Although z is
normally up and down.
But whatever the
next dimension is,
you can define a vector k that
goes into that third dimension.
Here I'll do it in a kind
of unconventional way.
I'll make k go in
that direction.
Although the standard
convention when you do it
in three dimensions is that k
is the up and down dimension.
But this by itself is
already pretty neat
because we can now represent any
vector through its components
and it's also going to
make the math much easier.