Vector triangle inequality - Mathematics, Engineering Video Lecture - Engineering Mathematics

In the last video, we showed
you the Cauchy-Schwarz
Inequality.
I think it's worth rewriting
because this is something
that's going to show up a lot.
It's a very useful tool.
And that just told us if I have
two vectors, x and y,
they're both members of Rn.
And they're both nonzero
vectors.
And that was an assumption we
had to make when we did the
proof, otherwise there was a
potential of dividing by one
of their magnitudes.
So that would've been
a big no-no.
But if we assume that they're
both nonzero, then we can say
that the absolute value of their
dot products is going to
be less than or equal
to the products of
their individual lengths.
So that's the length of vector x
and we defined that a couple
of videos ago.
And then this is the
length of vector y.
And of course, this is just a
regular number and then each
of these are just
regular numbers.
They're not vectors once
you take a length.
The length of a 50 dimensional
vector could just
be the number 3.
It's just a scalar value.
So this is just scalar
multiplication here.
And we also learned that the
only time that this inequality
turns into an equality is the
situation where x is equal to
some scalar multiple of y.
And so in some textbooks you'll
say-- and this has to
be a nonzero scalar multiple.
But that's a bit obvious.
I told you that x and
y are nonzero.
So if this was 0 then
x would be 0.
And we already said
that x is not 0.
But if you want to say there you
could say that you know c
also is going to be nonzero.
But that essentially
just falls out of
this information there.
But if this is the case and if
and only if this is the case,
then we can say that the
absolute value of the dot
product of the two vectors
is equal to the
product of their lengths.
Now, this is all just
a review of what I
did in the last video.
Now what else can we do
that's useful with it?
So let's just play around
a little bit.
I can't claim to be
experimenting, I know where
this is going to go.
Let's see what happens
if I were to take the
length of x plus y.
So I'm going to add the two
vectors and then take the
length of that vector squared.
Well we know from a couple of
videos ago that the length
squared can also be rewritten as
the dot product of a vector
with itself.
This right here, x
plus y, I know it
looks like two vectors.
But it's two vectors added
to each other.
So it's really a vector. x
plus y is a real vector.
I could graph x plus y.
So the length of x plus y
squared, I can rewrite it as
the dot product of that
vector with itself.
So x plus y dot x plus y.
And all of these are vectors.
These aren't just numbers.
And this is the dot product.
It's just not normal
multiplication.
But we saw two videos ago that
the dot product has the
distributive and the
associative and the
commutative properties just
like regular scalar
multiplication.
So you can kind of FOIL this out
if that's how you remember
multiplying your binomials.
Or I will think of it more as
just doing the distributive
property twice.
So this can be rewritten
as x dot x.
Actually, let me write it as
the distributive property
because that's sometimes not
obvious to a lot of people.
So let me write this x as a
yellow x and let me write this
whole term as x plus y.
So this right here can be
rewritten as x dot-- so this x
dot this x plus y.
And then it would be plus
this y dot-- I want
to just switch colors.
Plus this y dot x plus y.
It's good to see that when
you're multiplying these,
you're just applying the
distributive property.
All I did is I distributed this
term along each of these
terms into sum right here.
So then I got this.
And then I can distribute each
of these into the sum.
So then this becomes-- I'll be
careful with the colors-- x
dot x plus x dot y.
Maybe this was a little bit
overkill, but I think it's
good to see that this isn't
just some magic here.
And we're just using the exact
properties that we proved with
the dot product.
So that's that right there.
And then it's plus y dot x.
Plus this yellow y
dot the yellow x.
Sorry, dot the blue y.
So the magnitude or the length
of our vector x plus y squared
can be rewritten like this.
And I'll just switch
back to one color.
So this equals that and all of
that-- what does this equal?
This is equal to x dot x.
What's x dot x?
x dot x is just the magnitude.
So let me write this is just
equal to the magnitude
of our vector x.
I should stop using the
word magnitude.
The length of our vector
x squared.
And then I have two
terms here.
I have an x dot y
and a y dot x.
We know that x dot y and y dot
x are really the same thing.
We showed that order doesn't
matter when you take the dot
product, just like it doesn't
matter with regular
multiplication.
So these are really the same
terms. So we could write plus
2 times x dot y.
And then finally, we have that
last term sitting here.
We have this y dot y.
y dot y is the same thing
as the length of
our vector y squared.
Now, let's see if we can break
out our Cauchy-Schwarz
Inequality.
Or maybe Schwarz, I don't know
if I'm pronouncing it right.
But x dot y.
t We have the absolute value
of x dot y here.
But we know that just x dot y is
going to be-- it has to be
less than or equal to the
absolute value of x dot y.
Why is that?
Well this could be negative.
I could show you
examples of dot
products that are negative.
In fact, if x has all positive
terms and y has all negative
terms, you're going to have
a negative dot product.
So this could be negative
or it could be positive.
If it's positive the absolute
value-- their
equal to each other.
If this is negative, than this
absolute value is definitely
going to be greater than it.
We can add to the Cauchy-Schwarz
Inequality and
this is a bit obvious.
We could say look, we could add
a little x dot y is less
than or equal to the absolute
value of x dot y.
Which is less than or equal to
the length of x times the
length of y.
So x dot y is definite-- this,
the dot product of x with y is
definitely less than it's
absolute value of that.
Which is definitely less than
the lengths of those two
multiplied.
So if I rewrite this, this
statement right here is
definitely less than or equal
to this exact statement.
But if I replace these with the
lengths of the vectors.
So that is definitely less than
or equal to-- I'm just
rewriting this x squared and
I'll write the plus 2 there.
Plus 2.
But I want to make it very clear
what I'm replacing here.
And then I have the
plus length of my
vector y there squared.
Now this I'm saying, this is
definitely less than the
absolute value of x dot y.
Which is definitely less, by the
Cauchy-Schwarz Inequality,
definitely less than the product
of the two lengths.
So I'm just replacing this
with the product
of their two lengths.
So I'm going to put the length
of x times the length of y.
And since this is the
same as that, this
is the same as that.
But this is definitely
less than this.
This whole term has to be less
than this whole term.
Now let me just remind you
what we were doing.
I said that this thing that I
wrote over here, this is the
same as that.
So this thing up here, which is
the same as that, which is
less than that also.
So we can write that the
magnitude of x plus y squared
and not the magnitude, the
length of the vector x plus y
squared is less than
this whole thing
that I wrote out here.
Or less than or equal to.
Now, what is this thing?
Remember, I mean this might look
all fancy with my little
double lines around
everything.
But these are just numbers.
This length of x squared,
this is just a number.
Each of these are numbers and I
can just say hey, look, this
looks like a perfect
square to me.
This term on the right-hand side
is the exact same thing
as the length of x plus
the length of y.
Everything squared.
If you just squared this out
you'll get x squared plus 2
times the length of x times the
length of y plus y squared.
So our length of the vector x
plus y squared is less than or
equal to this quantity
over here.
And if we just take the square
root of both sides of this,
you get the length of our vector
x plus y is less than
or equal to the length of the
vector x by itself plus the
length of the vector y.
And we call this the triangle
inequality, which you might
have remembered from geometry.
Now why is it called the
triangle inequality?
Well you could imagine
each of these to be
separate side of a triangle.
In fact, let's draw it.
We can draw this in R2.
Let me turn my graph paper on.
Let me see where the
graphs show up.
If I turn my graph paper
on right there, maybe
I'll draw it here.
So let's draw my vector x.
So let's say that my vector x
look something like this.
Let's say that's my vector x.
It's a vector 2, 4.
So that's my vector x.
And let's say my vector y-- well
I'm just going to do it
head to tail because I'm
going to add the two.
So my vector y-- I'm going to do
it in nonstandard position.
Let's say it's look something
like-- let's say my vector y
looks something like this.
Draw it properly.
That's my vector y.
What does x plus y look like?
And remember, I can't
necessarily draw any two
vectors on a two-dimensional
space like this.
I'm just assuming that
these are in R2.
But this is just to
give you the idea.
So then this is their
sum, right?
You took from the tail of
x to the head of y.
So this vector right here
is the vector x plus y.
And that's why it's called
the triangle inequality.
It's just saying that look, this
thing is always going to
be less than or equal to-- or
the length of this thing is
always going to be less than or
equal to the length of this
thing plus the length
of this thing.
And that's kind of obvious
when you just learn
two-dimensional geometry.
That look, this is a much more
efficient way of getting from
this point to this point
than going out here and
then going out here.
And then, what's the case in
which this length is equal to
these lengths?
Well if you keep flattening this
triangle out and you go
to the extreme case where
maybe the vector
x looks like this.
And if the vector y is just
kind of going in the exact
same-- vector y is going in
the exact same direction,
maybe it's going a little
bit further.
This is vector x, this
is vector y.
Now x plus y will just
be this whole vector.
Now that whole thing
is x plus y.
And this is the case now where
you actually-- where the
triangle inequality turns
into an equality.
That's why that little
equal sign is there.
The extreme case where
essentially, x and y are collinear.
And why does that work out?
We can even go back to our
math and understand that.
So let me turn my graph off.
We can go back to
our math here.
If I go back to this point,
remember, right here I made
the statement, look, this thing
is definitely less than
this thing over here.
But what if I made
an assumption?
What if I said that x is equal
to some scalar times y?
And actually, I have to be
careful because just some
scalar times y-- remember, our
Cauchy-Schwarz Inequality said
that look, the inequality turns
into an equality if x is
some nonzero scalar of y.
And then we can apply this.
We can say that the absolute
value of x dot y is the same
as this over here.
But I don't have the absolute
value of x dot y here.
I don't know that this
is positive.
I can say definitively that this
is a positive quantity
because I took the absolute
value of it.
No absolute value here.
So the only way that I can
assure that this is a positive
quantity, that this is the same
thing as the absolute
value of x dot y is to enforce--
if I'm kind of going
to go down this road, is to
enforce that this term right
here, that c be positive.
Because if c is positive, then
x dot y, if x dot y then that
would be the same thing
as cy dot y.
Which we know is just the same
thing as c times the magnitude
of y squared.
And the only way that I can
ensure that this expression
right here is equal to the
absolute value of x dot y, the
only way I can assure this
is that c is positive.
If c is negative, then this is
going to be a negative number
while this is a positive.
So if I assume that this is
positive, then I can say that
x dot y is equal to the absolute
value of x dot y.
And since it's a scalar
multiple, then I could say
that that term is equal to, not
just less than or equal
to, the magnitude
of x's and y's.
So hopefully I'm not
confusing you.
So all I'm saying is, if I can
assume that x is some positive
scalar multiple of y,
that this wouldn't
be a less than sign.
Then I could say that x dot
y is the same thing as the
absolute value of x dot y
since this is positive.
And if it's the same thing as
the absolute value of x dot y
and it's some scalar multiple
of each other, than we could
go down this other route.
We could say that this
thing here-- I don't
want to get too messy.
We could say that this
is equal to that.
If this is equal to that, then
this would have become an
equal sign, not a less than
or equal to sign.
And then we would have had the
limiting case-- I don't want
to call it the limiting case.
But we could say that x plus y--
we would've done the same
work, but we would've had an
equal sign the whole way.
Would equal the length of x.
The length of x plus y would
equal the length of x plus the
length of y in the situation
where x is equal to some
positive scalar times y.
So c is greater than 0.
These two imply each other.
And we saw that geometrically.
I lost my axes here, but we see
that the only time that
the length of x plus y is equal
to the length of x plus
the length of y is when
they're collinear.
Over here this plus this is
clearly-- you can just
visually look at it-- longer
than this right there.
So you might be saying Sal,
once again, this linear
algebra's a little bit silly.
We learned the triangle
inequality in
eighth or ninth grade.
Why did you go through all of
this pain to redefine it?
And this is the interesting
thing.
What I just drew here and this
is what you learned in ninth
grade geometry.
This is just in R2.
This is just your Cartesian
coordinates, or I don't want
to use the word dimension too
much because we're going to
define that formally.
But this is kind of your
two-dimensional space
that's going on.
What's interesting or what's
useful about linear algebra
is, we've just defined the
triangle inequality for
arbitrarily large vectors,
or vectors that have an
arbitrarily high number
of components.
Each of these, these don't
have to be in R2.
This is true if we're in R100
where every vector has a
hundred components to it.
We've just defined some notion
of the triangle inequality.
We've abstracted well beyond
just our two-dimensional
Cartesian coordinates.
Well beyond even our three
dimensions to essentially, n
dimensional space.
And I haven't defined dimensions
yet, but I think
you're starting to appreciate
what they are.
But anyway, hopefully you
found that useful.
We can now take this result.
And actually, that result with
this result and define what
the notion of an angle between
two vectors are.
Once again, you know, on some
levels you're like well, why
do we have to define an angle?
Isn't an angle just-- isn't
that just an angle?
Well yeah, we know what an angle
is in two dimensions,
but what does an angle mean when
you abstract things to n
dimensions?
Or when you're in Rn.
So that's what we'll talk
about in the next video.