Some things are strictly a question of your point of view
For example, Copernicus said
Everyone has always looked at the universe from the point of view of the Earth
What would happen if we look at it from the point of view of the Sun?
You know what the result of that was
All heaven broke loose
Obviously, your point of view is not something to be taken lightly
And once you've chosen a point of view
you need to have some means of describing exactly where something is
and which way it is going
How can we do that?
Well there are many ways, for example,
chess players have a scheme of telling you
exactly where on the board each piece is and how it moves
And children playing the game of battleship
understand immediately how to describe
exactly where something is
And of course, it's not only chess players and children
who have to be able to do that
we scientists have to have some means of describing
exactly where something is
In the Northern Pacific Ocean, when there's a battleship involved here
it's probably not involved in child's play
This is the Alameda Coast Guard Center
on a strategic sliver of land in the San Francisco Bay
At here, the methods to locate a vessel
or to launch one at a moment's notice are strictly professional
From uneventful hours on watch
to times of crises on the high seas
United States Coast Guard does its duty
with a considerable variety of ways and means
Navigation charts accurate to the pinpoint
A deep familiarity with the Bay and surrounding waters
Sophisticated electronics
Keen tracking devices
Personnel beyond compare on North American shores
These are the resources of the US Coast Guard
And out of sight but not out of mind
in their powerful equation of talent, training and technology
there are the ever-ready tools of vector mathematics
A quantity that has both magnitude and direction
is a vector, represented by an arrow
The arrow's direction is the direction of the quantity
And the length of the arrow indicates its magnitude
Displacement, velocity and acceleration are all vectors
In equations, vectors are written in bold face
Ordinary quantities such as time and mass are scalars
In equations, scalars are written in italics
The magnitude of a vector is also a scalar
The same letter but in italics.
With vectors, even the most familiar mathematical operations acquired an entirely new meaning
For example, when two vectors are added together
the sum isn't merely a number, it's a new vector
Also, vectors can be subtracted from each other
and the result is another new vector
Of course, whether or not the Coast Guard uses classical scientific methods
their tools are up to the minutes and ready to be called upon
Irish Coffee, this is Sam. Please report on traffic. Over.
An officer receives the signal of distress
And no matter what the future may hold at the other end
He's well prepared
He has to be
In a normal course of duty
the Coast Guard rescues all manners of sea creatures
And as a matter off course
the Coast Guard even rescues those who seem better off on dry land
A crew of Sunday sailors on the good yacht Irish Coffee
It seems they've drifted off course
Way off course, to points entirely unknown
Likewise unknown, it's exactly how the captain and crew got themselves into this mess
In any case, the Coast Guard uses its own resources
And at the flick of a switch
they can swing into action
But wait a sec
this approach may not be necessary
And at least for the moment, from the crew's point of view
as long as the refreshments hold out
the situation is not yet a crisis on the good ship Irish Coffee
Malcolm Bogart and Percival Flynn
brave men who, on the high seas, take the courage from their leader Captain Duke
However, no matter whether the day is indeterminate or merely potential
it's never taken lightly by the United States Coast Guard
Irish Coffee, this is Sam from the Coast Guard, location and nature of distress, over.
This is the Irish Coffee, repeat, we're in trouble and we're in some place off the coast of San Francisco.
Irish Coffee, this is Sam from the Coast Guard, the quest is on Channel 13 and contact will be in 15 minutes, over.
Great, not only we are not sailors, we're not map readers, either
Where do we sit? It's got to be the blue part, right? Guess that's where we are
We're on it, we've got to be the blue one
Being careless to be lost at sea is no laughing matter
And an uneducated guess can make the situation perilous at best
Particularly, when these coordinates are on the San Francisco road map
Nonetheless, there's something to be learnt here
These lines form a rectangular grid
at least to the extent of the spherical surface of the Earth permits
Any point on this map can be located by determining its horizontal and vertical positions
For example, picture some point on the map marked with a red dot
Suppose the red dot has a horizontal coordinate x of -150km
And suppose it has a vertical coordinate y of -124km
Whatever the distance is
whether on land or sea or in the air
a coordinate system is vital to any map
Street maps of major cities usually feature grids
And like New York cities, each map has a system of coordinates on its own
About even with world-famous landmarks such as Central Park
Boston Common
or the White House on Pennsylvania Avenue
Maps can bewilder one who is unfamiliar with rectangular grids
Of course, to the mail carrier
fire fighter, delivery person to the police officer or taxi cab driver
that same grid is simply a way to get from one place to another
But to some, finding one's way around the world presents a challenge
A challenge such as that one met by a man
who actually understood the meaning of the word 'coordinates'
In the history of mathematics, René Descartes and Pierre de Fermat
wrote the opening chapter on coordinate systems
In the 17ᵗʰ century, these French fellows cleverly plotted the scheme
to connect geometry and algebra
At the same time, across the English Channel
John Wallis developed the theory along similar lines
Though Carl Friedrich Gauss would later coin the term
Wallis was the first to introduce the idea of the complex number
In his book Algebra, Wallis represented a complex number
'a+bi'
For example, by measuring the real part 'a'
along a horizontal axis
and the imaginary part 'b'
along a vertical axis
Even today, the numbers on a grid map can look complex, indeed
And if the paths of the city seem perplexing
consider the plight of strangers to the sea
Hey, that's it! Look! There's the plane.
Hello! There it is! C'mon!
Hurry up before it goes!
Getting lost without a good map is the easy part
getting found without one takes a little more work
Approaching a lost vessel
the Coast Guard has a number of different methods on hand
One of them is called "Triangulation"
Two different angles of radio reception are established by using directional antennas
And straight lines are extended until they intersect
Now if the Coast Guard were to use triangulation
and if the sea were as predictable as mathematics
the Irish Coffee would be just about here
That is, just about, here
if the Irish Coffee was stable enough to stay put
But that's too much to expect
Because, at the mercy of currents, vessels drift all over the sea
Currently then, knowing her location at one fixed time and place isn't enough
The Coast Guard needs to get the direction and the speed
of the Irish Coffee's drift
Which way?
How fast?
While the Captain obviously has a novel approach
he won't find such answers in the pages of Moby Dick
Flynn's smoke signals seem to be going nowhere
So, another distress signal probably can't do any more harm
This is the Irish Coffee. We're off the coast of San Francisco. Please come here, Coast Guard.
Now, if Bogart actually gave the Coast Guard a second position
the picture could look like this
Both positions are joined by an arrow
from position 1 to position 2
The arrow's direction shows which way the vessel has drifted
Its length shows how far
This arrow is called a displacement vector
With more information
the Coast Guard can estimate the path of the drifting vessel
by joining the displacement vectors tail-to-head
which is just vector addition
They can also estimate the drift velocity
by dividing the vectors by the time between signals
A vector can be multiplied by a scalar
This new vector has the same direction if the scalar is positive
But its magnitude can be different
Multiplying a vector by a negative scalar
reverses its direction
Of course, the modern Coast Guard rarely finds it necessary
to estimate the path of a drifting vessel
by sketching displacement and velocity vectors by hand
But nonetheless, when it comes to grasping
certain principles of navigation and mathematics
the value of the vector can't be overestimated
Nor even here, can its history be overlooked
Vector algebra was born in the 19ᵗʰ century
and its conceptual godfathers were William Rowan Hamilton, an Irish man
and a German by the name of Hermann Grassmann
For his part in the creation of this mathematics
Grassmann tried to develop an algebraic structure
on which geometry of any number of dimensions could be based
Although many viewed this work as too complicated at the time
the seeds beneath Hermann Grassmann's mathematics
stemmed from the potent intellect of Ancient Greece
After all, the parallelogram law of composition of forces
which Aristotle considered in a special case of a rectangle
Here's an example of adding vectors
And one way or another, that idea has been considered
with the greatest care ever since
Isaac Newton thought enough of the parallelogram law of forces
to incorporate it into his magnificent work: The Principia
But even then, it would take a couple of hundred years
and William Hamilton to see the scope and versatility of the vector
In his attempt to find the mathematical way
to interpret rotation in space in physical terms
Hamilton saw a three-dimensional analogue of complex numbers
Instead, he found an algebraic four-dimensional object
which he called "Quaternions"
A quaternion has a vector part, which is three-dimensional
and a one-dimensional scalar part
Though it would take a bit of modern refinement in the vector and the scalar
William Hamilton had devised an enormously valuable mode
of expressing Newton's mechanics
James Clerk Maxwell, a pioneer in electricity and magnetism
saw the theoretical brilliance of Hamilton's ideas
But even so, they weren't generally accepted
To be effective, content and form have to work together
And while Hamilton's theory was simply brilliant
its form was too complicated to survive
In essence however, the theory did survive
In the 19ᵗʰ century, a professor of Yale, Josiah Willard Gibbs
took original ideas from Hamilton and Grassmann
and applied them to theoretical physics
However, reluctant to take credit for ideas that weren't his to begin with
Gibbs waited two decades before permitting E. B. Wilson
to reconstruct his notes into book form
Finally, with publication in 1901
vectors sailed around the globe
Until, like the sound of music
vectors could be heard from just about everywhere there's a meeting of the minds
Of course, vector algebra, like music notation, takes some practice
And even aboard the Irish Coffee
a little exercise couldn't make this situation much worse
And in any case, vectors, unlike the lyrics of certain songs
are quite versatile
A vector can be multiplied by another vector
The dot product of a and b is a scalar
that measures the tendency the two vectors to point in the same direction
a·b is equal to the length of a
times the length of b
times the cosine of the angle between them
If a and b are perpendicular, their dot product is zero
And the dot product of a vector with itself
is just the square of its length
Moving right along
The cross product of two vectors is a new vector
perpendicular to the plane of the two original vectors
Its length is the area of the parallelogram formed by original vectors
Its direction is determined by the right hand rule
The cross product measures the tendency of two vectors
to be perpendicular
And it's one of the more effective tools
for describing spinning or rotating objects
Vector algebra makes use of two perpendicular unit vectors
i and j
The little hat over the vector means that it has length 1
The vector from the origin to the point with coordinates (x,y)
is the sum of two perpendicular vectors
a horizontal vector xi and a vertical vector yj
With the help of i and j
adding and multiplying vectors can be accomplished by ordinary algebra
In three-dimensional space, a third unit vector k is used
k is perpendicular to the plane of i and j
A three-dimensional vector is written
xi+yj+zk
A moving song is one thing
the vessel moving right along is quite another
especially when she's drifting at sea
Remember, before the first distress signal
the wind was calm, the water smooth
the day perfect for a little good beer drink
But conditions change
and the Irish Coffee isn't where, or what, she used to be
Again, the wind at sea, like the captain's crew
is a force to be reckoned with
Wind velocity has both speed and direction
which means it's a vector quantity
So is the velocity of flowing water
And so now, drifting with the wind and the water
could getting nowhere fast
These men are at a loss
And finally, they realized
That's it.
Of course, if the captain were mathematically inclined
he'd try to figure out precisely how to help the Coast Guard
He'd determine his vessel's position
perhaps, 150km west
and 124km south of the Alameda Coast Guard base
And of course, he'd come up with an estimate of wind velocity
say, 40km/h out of the south
and the he'd calculate that his vessel
is being pushed along at an average speed of 2km/h due north
However, as the men of the Irish Coffee have come to realize
Captain Duke, while he is called many things
cannot be called a mathematician
That's no criticism
But when it comes to vectors, the coordinates
to whatever may be used for a rescue
it just doesn't have what it takes
Unless that is, it takes a vivid imagination
The way Captain Duke sees it
the US Coast Guard wouldn't get very far without him
So with his help, to say nothing of coordinates
wind velocity and probable drift
and a helicopter thrown in for good measure
things can really take off
In Captain Duke's imagination
the pilot has enough fuel and maximum aerial speed at 125km/h
to remain in the air for hours
That means 2 hours out, 2 hours back
and 250km each way, no more
Can the helicopter reach the Irish Coffee?
and surely important
Can the pilot make it back?
Wind carries the copter due north, at 40km/h
80km due north in 2 hours
So, if the pilot were to fly along one of these vectors
he'd wind up just out of range
And even with Captain Duke's courage
he'd fly no farther
because he couldn't count on the steady wind to help him
on his way back to base
Still, the situation isn't as bad as it looks in Captain Duke's worst scenario
Why?
Because there, the Irish Coffee drifts as well
Heading north at 2km/h
And in 2 hours, she'll drift 4km due north
which would place her at the edge of the chauffeur's range
And accounting for aerial speed
wind speed and drift velocity
the pilot could calculate the right heading
and reach the Irish Coffee
Even in the Captain's fantasy
is that possible?
Yes. And in reality,
the answer should be obvious by now
The copter's path, and its displacement by the wind
combine or add as vectors
As simple as they are
such calculations would be vital if the US Coast Guard
read Captain Duke's mind
Instead, they went along with a plan of their own
We have a vessel to save southwest of the Fillmore...
Actually, the Irish Coffee is a few km south of the Farallon Islands
near the San Francisco Bay
And almost within shouting distance
So soon enough, captain and crew will be, in good hands
And long before their galley runs out of refreshments
they'll be back on dry land
And until they learn the lessons on seafaring
to say nothing of vectors
that's exactly where they belong
Before Copernicus, the center of the Earth
was the center of the universe
At the Aristotelian world
the very idea of place had no meaning
except where something was
with respect to the center of the Earth
Then Copernicus comes along
and thus affects the routine mathematical operation
Writing the equations of the orbits of the planets
in a different coordinate system
And the world is never the same again
But that's misleading
It makes it sounds as if the important thing
is to have the right coordinate system
And what we learn from all that
was really exactly the opposite
That all coordinate systems are equally good
Copernicus says the origin should be at the Sun
the Coast Guard says the origin should be at its coordinating station
And they are both equally right
In fact, we can say it in a way that's much more profound
The idea is that the laws of physics
are exactly the same everywhere
Newton's laws work as well in the Crab Nebula
as they do in Kansas City
And because we believe that
we need a way of expressing those laws
that works equally well in all coordinate systems
And that way, is by means, of vectors
The idea of a vector is disconcerting
because a vector has a size and a direction
but it has no place
But that's exactly what we need
in order to express laws that are the same
in every place
In fact, next time
we'll see a vector equation
which lies at the heart of our understanding of the world
I'll see you then
Subtitle created by Tran Nguyen Phuong Thanh - 2014