Welcome back.
In the last video, I at the end of the video, like I
always do in the attempt to confuse you, I told you that
if I had two vectors-- And let me just make up some new ones,
so I can draw them visually in a second or two.
Let's call the first vector a.
Let me do a different color.
This toothpaste color is getting monotonous.
Let me do something that looks relaxing.
Let's call a first vector a and, I don't know, let's make
it interesting, let me say it's minus 3 times the unit
vector i plus 2 times the unit vector j.
And then I have vector b.
And that is equal to, 2i, so two times the unit vector i.
Plus, 4 times the unit vector j.
In the last video I said, well, the whole reason why
this unit vector notation is even -- Well, one of the
reasons, we'll see that there many reasons why it's useful.
One of the really cool things about it is, before when we
added vectors, we would put them head to tails, and then
draw it visually, and then we had this new vector.
And we really had no way of expressing it
without drawing it.
But when we write things as multiples of the unit vectors.
We don't have to draw it.
And it's actually very easy to add vectors.
And how do we do it?
We just add the x components, and we add the y components.
So we said that these two vectors, a plus b, these
little weird arrows on top, that's just saying that those
are vectors.
That's equals.
So it's minus 3, plus 2i, and I'm going to arbitrarily
switch colors, because it's getting monotonous.
Plus 2 plus 4j.
We just added the x components, or the
multiples of i.
And we added the y components, or just the multiples of j.
Because i was the unit vector in the x direction, and j was
the unit vector in the y direction.
And we get, what's minus 3 plus 2?
That's minus 1.
We get minus 1i.
That could just be minus i.
But I'll write the 1 because we're just getting warmed up
with unit vectors.
So minus 1i plus 6j.
And when I did that, you might say, well, Sal, I can't just
take your word for it.
Because you seem not someone who
should be believed blindly.
So I think that's a valid opinion to have. So I will
show you that this works, by adding the vectors visually.
So let's draw it.
And I think this will give you a little better sense of unit
vectors generally.
Let me draw the axes.
So that's my y-axis.
Let me draw my x-axis.
I have to make sure have enough space to draw the unit
vectors that we've drawn, or to draw the
vectors that we've drawn.
Just to show that the axes go on forever, I have to draw
that arrow.
All right, so let's say this is 1, 2, 3.
This is 1, 2, 3, 4.
And I draw 1, 2, 3, 4, 5, 6.
I think we should be able to now add them.
I didn't have to waste all this space down here.
So let's just first draw the vectors, minus 3i plus 2j.
So minus 3i, just this right here, is going to be a vector
that looks something like this.
So it's just minus 3 times the x vector, so
it'll go to the left.
Because i is 1 in the positive direction.
If we put a negative there, it flips it over.
Let me use a different color.
So this is minus 3i, and then plus 2j.
So plus 2j looks like this.
If we were to add those two vectors visually, we can put
them head to tails.
And the way we can do that, we can either shift this vector
up like this, and draw it up here.
Or we could shift this vector, and put its tail
its vector's head.
But either way, let's shift this one up.
So if we shifted up like that.
Remember, we're just doing the head to tails, visual addition
method of vectors.
So I just put this tail to this head.
And what do we get?
So vector a will look like this, and I'm going to do it
in the same color as vector a because I have a feeling that
this diagram might get complicated.
Well, I wanted to use the line tool.
OK, so this is vector a.
That's what vector a looks like.
And so we worked backwards.
I gave you the x component and the y component.
And then I added them together by doing the head to tails
method, and so this is what vector a would look like.
And, instead of drawing it, a very easy representation is
exactly what we did up here, a unit vector notation.
And what's vector b look like?
So it's 2i-- I'm going to do a completely different color.
It's 2i, so it's this vector.
2 times unit vector i.
That's this.
Plus 4j, 1, 2, 3, 4.
So it looks like this.
And let's take this one and shift it over to the left, so
we can put its tail to the vector's head, so it would
look like this.
So vector b will look -- I'll do it in red.
And I'll use a line tool.
Vector b looks like this.
I just put its components head to tails, and that's how
I got vector b.
And if I were to add them visually.
I would do it the same way that I added its components.
I would put the tail of one vector to the head of the
other, and see if you get the resulting vector.
So you could do it either way.
Let's shift this a vector.
Let's shift it in this direction.
Remember, vectors, we're just giving the
magnitude of direction.
We're not necessarily giving a starting point.
So you can shift them.
You just can't change their orientation or their
magnitudes.
And that's actually how you add them, you shift them, and
put them head to tails.
That's when you add them visually.
Let's put that a vector up here.
So if we have the a vector, it looks something like this.
And I want it to work out right.
So the a vector looks something like that.
And remember, all I did was I took the same vector, and I
just shifted it.
So that it can start at the head.
So its tail can start at the head of the b vector.
I just shifted the a vector, so this is still the a vector.
By moving the vector around, you
haven't changed the vector.
I would only change the vector, if I scaled it, if I
made it bigger or smaller, if I changed its orientation.
And so visually, this is b, this is a, so if I add a to b,
the resulting vector, going head to tails-- i'll do it in
this green color --would look like this.
It would look like that.
So here we took all this trouble, and I had to draw
these straight lines to visually
add these two vectors.
This green vector is a plus b.
Let's see if this green vector is the same
thing that we got here.
Let's see if it's the same thing as this.
So we got negative 1 times i, so negative 1 is here.
And then we have 6j.
Let me do it in another color.
6j would look like this.
6j looks like that.
You put them heads to tails.
And it would be something like this.
And that is the green vector.
And actually, just so you know, I know it didn't line up
perfectly, and that's because I'm not drawing neatly, but
these two points should actually be here if I were to
have drawn this better.
But I know this is very confusing, I
had all these colors.
But the whole point of it is, I wanted to show that you
could visually draws vectors, and then shift them around,
and then put them heads to tails.
And then get the resulting vector.
That's one way to add vectors, there's still no way to
analytically represent it.
Or you could just write any vector as its x and y
components, and then the sum of the vectors is just going
to be the sum of the x's and the sum of the y's.
And that's a much cleaner, and a much easier, and much less
prone to error, way of adding or subtracting two vectors.
So hopefully that was convincing.
That a plus b really is this vector.
If it wasn't, I'm sorry.
And I hope I didn't confuse you more.
But now that we have this out of the way, and hopefully
you're convinced that unit vector notation is useful.
We can move on and maybe try to do some of our old
projectile motion problems using this notation.
And maybe it'll let us to do a little bit of
extra stuff with it.
See you soon.