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Welcome back.

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In the last video, I at the
end of the video, like I

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always do in the attempt to
confuse you, I told you that

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if I had two vectors-- And let
me just make up some new ones,

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so I can draw them visually
in a second or two.

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Let's call the first vector a.

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Let me do a different color.

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This toothpaste color is
getting monotonous.

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Let me do something that
looks relaxing.

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Let's call a first vector a and,
I don't know, let's make

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it interesting, let me say it's
minus 3 times the unit

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vector i plus 2 times
the unit vector j.

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And then I have vector b.

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And that is equal to, 2i, so two
times the unit vector i.

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Plus, 4 times the
unit vector j.

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In the last video I said, well,
the whole reason why

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this unit vector notation is
even -- Well, one of the

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reasons, we'll see that there
many reasons why it's useful.

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One of the really cool things
about it is, before when we

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added vectors, we would put them
head to tails, and then

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draw it visually, and then
we had this new vector.

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And we really had no way
of expressing it

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without drawing it.

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But when we write things as
multiples of the unit vectors.

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We don't have to draw it.

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And it's actually very
easy to add vectors.

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And how do we do it?

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We just add the x components,
and we add the y components.

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So we said that these two
vectors, a plus b, these

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little weird arrows on top,
that's just saying that those

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are vectors.

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That's equals.

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So it's minus 3, plus 2i, and
I'm going to arbitrarily

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switch colors, because it's
getting monotonous.

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Plus 2 plus 4j.

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We just added the x
components, or the

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multiples of i.

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And we added the y components,
or just the multiples of j.

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Because i was the unit vector in
the x direction, and j was

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the unit vector in
the y direction.

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And we get, what's
minus 3 plus 2?

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That's minus 1.

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We get minus 1i.

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That could just be minus i.

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But I'll write the 1 because
we're just getting warmed up

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with unit vectors.

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So minus 1i plus 6j.

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And when I did that, you might
say, well, Sal, I can't just

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take your word for it.

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Because you seem
not someone who

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should be believed blindly.

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So I think that's a valid
opinion to have. So I will

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show you that this works, by
adding the vectors visually.

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So let's draw it.

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And I think this will give you
a little better sense of unit

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vectors generally.

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Let me draw the axes.

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So that's my y-axis.

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Let me draw my x-axis.

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I have to make sure have enough
space to draw the unit

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vectors that we've drawn,
or to draw the

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vectors that we've drawn.

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Just to show that the axes go
on forever, I have to draw

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that arrow.

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All right, so let's say
this is 1, 2, 3.

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This is 1, 2, 3, 4.

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And I draw 1, 2, 3, 4, 5, 6.

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I think we should be able
to now add them.

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I didn't have to waste all
this space down here.

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So let's just first draw the
vectors, minus 3i plus 2j.

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So minus 3i, just this right
here, is going to be a vector

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that looks something
like this.

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So it's just minus 3 times
the x vector, so

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it'll go to the left.

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Because i is 1 in the
positive direction.

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If we put a negative there,
it flips it over.

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Let me use a different color.

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So this is minus 3i,
and then plus 2j.

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So plus 2j looks like this.

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If we were to add those two
vectors visually, we can put

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them head to tails.

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And the way we can do that, we
can either shift this vector

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up like this, and
draw it up here.

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Or we could shift this vector,
and put its tail

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its vector's head.

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But either way, let's
shift this one up.

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So if we shifted up like that.

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Remember, we're just doing the
head to tails, visual addition

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method of vectors.

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So I just put this tail
to this head.

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And what do we get?

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So vector a will look like this,
and I'm going to do it

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in the same color as vector a
because I have a feeling that

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this diagram might
get complicated.

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Well, I wanted to use
the line tool.

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OK, so this is vector a.

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That's what vector
a looks like.

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And so we worked backwards.

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I gave you the x component
and the y component.

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And then I added them together
by doing the head to tails

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method, and so this is what
vector a would look like.

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And, instead of drawing it, a
very easy representation is

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exactly what we did up here,
a unit vector notation.

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And what's vector b look like?

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So it's 2i-- I'm going to do a
completely different color.

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It's 2i, so it's this vector.

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2 times unit vector i.

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That's this.

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Plus 4j, 1, 2, 3, 4.

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So it looks like this.

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And let's take this one and
shift it over to the left, so

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we can put its tail to the
vector's head, so it would

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look like this.

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So vector b will look --
I'll do it in red.

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And I'll use a line tool.

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Vector b looks like this.

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I just put its components head
to tails, and that's how

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I got vector b.

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And if I were to add
them visually.

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I would do it the same way that
I added its components.

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I would put the tail of one
vector to the head of the

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other, and see if you get
the resulting vector.

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So you could do it either way.

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Let's shift this a vector.

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Let's shift it in
this direction.

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Remember, vectors, we're
just giving the

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magnitude of direction.

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We're not necessarily giving
a starting point.

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So you can shift them.

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You just can't change their
orientation or their

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magnitudes.

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And that's actually how you add
them, you shift them, and

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put them head to tails.

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That's when you add
them visually.

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Let's put that a
vector up here.

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So if we have the a vector, it
looks something like this.

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And I want it to
work out right.

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So the a vector looks
something like that.

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And remember, all I did was I
took the same vector, and I

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just shifted it.

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So that it can start
at the head.

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So its tail can start at the
head of the b vector.

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I just shifted the a vector, so
this is still the a vector.

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By moving the vector
around, you

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haven't changed the vector.

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I would only change the vector,
if I scaled it, if I

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made it bigger or smaller, if
I changed its orientation.

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And so visually, this is b, this
is a, so if I add a to b,

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the resulting vector, going head
to tails-- i'll do it in

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this green color --would
look like this.

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It would look like that.

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So here we took all this
trouble, and I had to draw

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these straight lines
to visually

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add these two vectors.

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This green vector is a plus b.

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Let's see if this green
vector is the same

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thing that we got here.

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Let's see if it's the
same thing as this.

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So we got negative 1 times
i, so negative 1 is here.

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And then we have 6j.

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Let me do it in another color.

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6j would look like this.

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6j looks like that.

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You put them heads to tails.

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And it would be something
like this.

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And that is the green vector.

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And actually, just so you know,
I know it didn't line up

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perfectly, and that's because
I'm not drawing neatly, but

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these two points should actually
be here if I were to

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have drawn this better.

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But I know this is
very confusing, I

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had all these colors.

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But the whole point of it is,
I wanted to show that you

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could visually draws vectors,
and then shift them around,

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and then put them
heads to tails.

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And then get the resulting
vector.

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That's one way to add vectors,
there's still no way to

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analytically represent it.

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Or you could just write any
vector as its x and y

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components, and then the sum of
the vectors is just going

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to be the sum of the x's
and the sum of the y's.

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And that's a much cleaner, and
a much easier, and much less

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prone to error, way of adding
or subtracting two vectors.

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So hopefully that
was convincing.

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That a plus b really
is this vector.

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If it wasn't, I'm sorry.

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And I hope I didn't
confuse you more.

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But now that we have this out
of the way, and hopefully

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you're convinced that unit
vector notation is useful.

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We can move on and maybe try
to do some of our old

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projectile motion problems
using this notation.

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And maybe it'll let us
to do a little bit of

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extra stuff with it.

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See you soon.

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