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Some things are strictly a question of your point of view
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For example, Copernicus said
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Everyone has always looked at the universe
from the point of view of the Earth
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What would happen if we look at it
from the point of view of the Sun?
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You know what the result of that was
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All heaven broke loose
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Obviously, your point of view is not something
to be taken lightly
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And once you've chosen a point of view
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you need to have some means of describing
exactly where something is
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and which way it is going
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How can we do that?
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Well there are many ways, for example,
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chess players have a scheme of telling you
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exactly where on the board each piece is and how it moves
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And children playing the game of battleship
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understand immediately how to describe
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exactly where something is
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And of course, it's not only chess players and children
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who have to be able to do that
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we scientists have to have some means of describing
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exactly where something is
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In the Northern Pacific Ocean,
when there's a battleship involved here
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it's probably not involved in child's play
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This is the Alameda Coast Guard Center
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on a strategic sliver of land in the San Francisco Bay
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At here, the methods to locate a vessel
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or to launch one at a moment's notice
are strictly professional
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From uneventful hours on watch
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to times of crises on the high seas
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United States Coast Guard does its duty
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with a considerable variety of ways and means
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Navigation charts accurate to the pinpoint
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A deep familiarity with the Bay and surrounding waters
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Sophisticated electronics
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Keen tracking devices
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Personnel beyond compare on North American shores
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These are the resources of the US Coast Guard
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And out of sight but not out of mind
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in their powerful equation of talent,
training and technology
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there are the ever-ready tools of vector mathematics
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A quantity that has both magnitude and direction
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is a vector, represented by an arrow
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The arrow's direction is the direction of the quantity
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And the length of the arrow indicates its magnitude
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Displacement, velocity and acceleration are all vectors
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In equations, vectors are written in bold face
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Ordinary quantities such as time and mass are scalars
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In equations, scalars are written in italics
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The magnitude of a vector is also a scalar
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The same letter but in italics.
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With vectors, even the most familiar mathematical
operations acquired an entirely new meaning
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For example, when two vectors are added together
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the sum isn't merely a number, it's a new vector
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Also, vectors can be subtracted from each other
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and the result is another new vector
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Of course, whether or not the Coast Guard
uses classical scientific methods
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their tools are up to the minutes and
ready to be called upon
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Irish Coffee, this is Sam. Please report on traffic. Over.
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An officer receives the signal of distress
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And no matter what the future may hold at the other end
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He's well prepared
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He has to be
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In a normal course of duty
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the Coast Guard rescues all manners of sea creatures
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And as a matter off course
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the Coast Guard even rescues those
who seem better off on dry land
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A crew of Sunday sailors on the good yacht Irish Coffee
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It seems they've drifted off course
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Way off course, to points entirely unknown
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Likewise unknown, it's exactly how the captain
and crew got themselves into this mess
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In any case, the Coast Guard uses its own resources
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And at the flick of a switch
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they can swing into action
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But wait a sec
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this approach may not be necessary
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And at least for the moment, from the crew's point of view
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as long as the refreshments hold out
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the situation is not yet a crisis
on the good ship Irish Coffee
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Malcolm Bogart and Percival Flynn
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brave men who, on the high seas, take the courage
from their leader Captain Duke
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However, no matter whether the day is
indeterminate or merely potential
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it's never taken lightly by the United States Coast Guard
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Irish Coffee, this is Sam from the Coast Guard,
location and nature of distress, over.
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This is the Irish Coffee, repeat, we're in trouble and
we're in some place off the coast of San Francisco.
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Irish Coffee, this is Sam from the Coast Guard, the quest
is on Channel 13 and contact will be in 15 minutes, over.
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Great, not only we are not sailors,
we're not map readers, either
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Where do we sit? It's got to be the blue part, right?
Guess that's where we are
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We're on it, we've got to be the blue one
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Being careless to be lost at sea is no laughing matter
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And an uneducated guess can make the situation
perilous at best
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Particularly, when these coordinates are on
the San Francisco road map
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Nonetheless, there's something to be learnt here
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These lines form a rectangular grid
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at least to the extent of the spherical surface
of the Earth permits
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Any point on this map can be located by determining
its horizontal and vertical positions
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For example, picture some point on the map
marked with a red dot
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Suppose the red dot has a
horizontal coordinate x of -150km
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And suppose it has a vertical coordinate y of -124km
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Whatever the distance is
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whether on land or sea or in the air
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a coordinate system is vital to any map
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Street maps of major cities usually feature grids
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And like New York cities, each map has a system
of coordinates on its own
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About even with world-famous landmarks
such as Central Park
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Boston Common
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or the White House on Pennsylvania Avenue
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Maps can bewilder one who is unfamiliar
with rectangular grids
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Of course, to the mail carrier
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fire fighter, delivery person to the police officer
or taxi cab driver
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that same grid is simply a way
to get from one place to another
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But to some, finding one's way around the world
presents a challenge
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A challenge such as that one met by a man
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who actually understood the meaning
of the word 'coordinates'
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In the history of mathematics,
René Descartes and Pierre de Fermat
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wrote the opening chapter on coordinate systems
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In the 17ᵗʰ century, these French fellows
cleverly plotted the scheme
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to connect geometry and algebra
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At the same time, across the English Channel
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John Wallis developed the theory along similar lines
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Though Carl Friedrich Gauss would later coin the term
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Wallis was the first to introduce the idea
of the complex number
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In his book Algebra, Wallis represented a complex number
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'a+bi'
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For example, by measuring the real part 'a'
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along a horizontal axis
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and the imaginary part 'b'
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along a vertical axis
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Even today, the numbers on a grid map
can look complex, indeed
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And if the paths of the city seem perplexing
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consider the plight of strangers to the sea
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Hey, that's it! Look! There's the plane.
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Hello! There it is! C'mon!
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Hurry up before it goes!
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Getting lost without a good map is the easy part
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getting found without one takes a little more work
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Approaching a lost vessel
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the Coast Guard has a number of
different methods on hand
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One of them is called "Triangulation"
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Two different angles of radio reception are
established by using directional antennas
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And straight lines are extended until they intersect
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Now if the Coast Guard were to use triangulation
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and if the sea were as predictable as mathematics
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the Irish Coffee would be just about here
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That is, just about, here
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if the Irish Coffee was stable enough to stay put
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But that's too much to expect
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Because, at the mercy of currents,
vessels drift all over the sea
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Currently then, knowing her location at one fixed
time and place isn't enough
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The Coast Guard needs to get the direction and the speed
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of the Irish Coffee's drift
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Which way?
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How fast?
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While the Captain obviously has a novel approach
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he won't find such answers in the pages of Moby Dick
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Flynn's smoke signals seem to be going nowhere
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So, another distress signal probably
can't do any more harm
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This is the Irish Coffee. We're off the coast of San Francisco.
Please come here, Coast Guard.
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Now, if Bogart actually gave the Coast Guard
a second position
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the picture could look like this
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Both positions are joined by an arrow
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from position 1 to position 2
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The arrow's direction shows which way
the vessel has drifted
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Its length shows how far
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This arrow is called a displacement vector
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With more information
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the Coast Guard can estimate the path
of the drifting vessel
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by joining the displacement vectors tail-to-head
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which is just vector addition
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They can also estimate the drift velocity
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by dividing the vectors by the time between signals
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A vector can be multiplied by a scalar
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This new vector has the same direction
if the scalar is positive
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But its magnitude can be different
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Multiplying a vector by a negative scalar
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reverses its direction
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Of course, the modern Coast Guard
rarely finds it necessary
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to estimate the path of a drifting vessel
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by sketching displacement and velocity vectors by hand
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But nonetheless, when it comes to grasping
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certain principles of navigation and mathematics
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the value of the vector can't be overestimated
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Nor even here, can its history be overlooked
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Vector algebra was born in the 19ᵗʰ century
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and its conceptual godfathers were
William Rowan Hamilton, an Irish man
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and a German by the name of Hermann Grassmann
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For his part in the creation of this mathematics
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Grassmann tried to develop an algebraic structure
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on which geometry of any number of dimensions
could be based
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Although many viewed this work as
too complicated at the time
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the seeds beneath Hermann Grassmann's mathematics
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stemmed from the potent intellect of Ancient Greece
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After all, the parallelogram law of composition of forces
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which Aristotle considered in
a special case of a rectangle
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Here's an example of adding vectors
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And one way or another, that idea has been considered
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with the greatest care ever since
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Isaac Newton thought enough of the
parallelogram law of forces
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to incorporate it into his magnificent work: The Principia
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00:15:01,400 --> 00:15:04,660
But even then, it would take a couple of hundred years
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00:15:04,660 --> 00:15:09,940
and William Hamilton to see the scope and
versatility of the vector
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00:15:09,940 --> 00:15:12,000
In his attempt to find the mathematical way
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00:15:12,000 --> 00:15:15,500
to interpret rotation in space in physical terms
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00:15:15,500 --> 00:15:19,360
Hamilton saw a three-dimensional analogue
of complex numbers
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Instead, he found an algebraic four-dimensional object
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which he called "Quaternions"
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A quaternion has a vector part, which is three-dimensional
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and a one-dimensional scalar part
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Though it would take a bit of modern refinement
in the vector and the scalar
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00:15:40,220 --> 00:15:43,780
William Hamilton had devised
an enormously valuable mode
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00:15:43,780 --> 00:15:46,220
of expressing Newton's mechanics
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00:15:47,740 --> 00:15:52,560
James Clerk Maxwell, a pioneer in
electricity and magnetism
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00:15:52,560 --> 00:15:55,880
saw the theoretical brilliance of Hamilton's ideas
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But even so, they weren't generally accepted
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00:15:59,480 --> 00:16:04,020
To be effective, content and form have to work together
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00:16:04,020 --> 00:16:07,220
And while Hamilton's theory was simply brilliant
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00:16:07,220 --> 00:16:10,800
its form was too complicated to survive
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00:16:10,800 --> 00:16:14,200
In essence however, the theory did survive
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00:16:14,200 --> 00:16:19,020
In the 19ᵗʰ century, a professor of Yale,
Josiah Willard Gibbs
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00:16:19,020 --> 00:16:22,140
took original ideas from Hamilton and Grassmann
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and applied them to theoretical physics
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00:16:25,420 --> 00:16:30,520
However, reluctant to take credit for ideas
that weren't his to begin with
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00:16:30,520 --> 00:16:33,980
Gibbs waited two decades before permitting E. B. Wilson
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00:16:33,980 --> 00:16:37,280
to reconstruct his notes into book form
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Finally, with publication in 1901
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00:16:40,740 --> 00:16:43,460
vectors sailed around the globe
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Until, like the sound of music
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00:16:46,300 --> 00:16:50,980
vectors could be heard from just about everywhere
there's a meeting of the minds
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00:16:55,320 --> 00:17:00,240
Of course, vector algebra, like music notation,
takes some practice
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And even aboard the Irish Coffee
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a little exercise couldn't make this situation much worse
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And in any case, vectors, unlike the lyrics of certain songs
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are quite versatile
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A vector can be multiplied by another vector
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00:17:19,640 --> 00:17:22,180
The dot product of a and b is a scalar
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that measures the tendency the two vectors
to point in the same direction
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00:17:27,420 --> 00:17:33,020
a·b is equal to the length of a
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times the length of b
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00:17:35,080 --> 00:17:38,380
times the cosine of the angle between them
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00:17:42,680 --> 00:17:47,220
If a and b are perpendicular, their dot product is zero
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And the dot product of a vector with itself
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is just the square of its length
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Moving right along
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The cross product of two vectors is a new vector
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00:18:13,980 --> 00:18:18,400
perpendicular to the plane of the two original vectors
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00:18:18,400 --> 00:18:24,540
Its length is the area of the parallelogram
formed by original vectors
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00:18:31,820 --> 00:18:36,220
Its direction is determined by the right hand rule
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00:18:36,220 --> 00:18:39,920
The cross product measures the tendency of two vectors
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00:18:39,920 --> 00:18:41,860
to be perpendicular
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And it's one of the more effective tools
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00:18:44,640 --> 00:18:49,560
for describing spinning or rotating objects
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00:19:03,880 --> 00:19:08,360
Vector algebra makes use of
two perpendicular unit vectors
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00:19:08,360 --> 00:19:10,540
i and j
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00:19:10,540 --> 00:19:14,980
The little hat over the vector means that it has length 1
254
00:19:18,660 --> 00:19:22,960
The vector from the origin to the point
with coordinates (x,y)
255
00:19:22,960 --> 00:19:26,080
is the sum of two perpendicular vectors
256
00:19:26,080 --> 00:19:31,820
a horizontal vector xi and a vertical vector yj
257
00:19:39,460 --> 00:19:41,760
With the help of i and j
258
00:19:41,760 --> 00:19:47,260
adding and multiplying vectors can be
accomplished by ordinary algebra
259
00:20:15,060 --> 00:20:20,960
In three-dimensional space, a third unit vector k is used
260
00:20:20,960 --> 00:20:24,800
k is perpendicular to the plane of i and j
261
00:20:26,100 --> 00:20:28,760
A three-dimensional vector is written
262
00:20:28,760 --> 00:20:39,680
xi+yj+zk
263
00:20:52,340 --> 00:20:54,300
A moving song is one thing
264
00:20:54,300 --> 00:20:56,940
the vessel moving right along is quite another
265
00:20:56,940 --> 00:20:59,700
especially when she's drifting at sea
266
00:21:00,500 --> 00:21:03,500
Remember, before the first distress signal
267
00:21:03,500 --> 00:21:06,220
the wind was calm, the water smooth
268
00:21:06,220 --> 00:21:09,920
the day perfect for a little good beer drink
269
00:21:09,920 --> 00:21:12,640
But conditions change
270
00:21:12,640 --> 00:21:17,280
and the Irish Coffee isn't where, or what, she used to be
271
00:21:19,540 --> 00:21:22,880
Again, the wind at sea, like the captain's crew
272
00:21:22,880 --> 00:21:25,120
is a force to be reckoned with
273
00:21:25,120 --> 00:21:28,100
Wind velocity has both speed and direction
274
00:21:28,100 --> 00:21:31,360
which means it's a vector quantity
275
00:21:31,360 --> 00:21:34,460
So is the velocity of flowing water
276
00:21:34,460 --> 00:21:38,500
And so now, drifting with the wind and the water
277
00:21:38,500 --> 00:21:40,940
could getting nowhere fast
278
00:21:40,940 --> 00:21:43,420
These men are at a loss
279
00:21:48,700 --> 00:21:51,300
And finally, they realized
280
00:21:51,300 --> 00:21:53,040
That's it.
281
00:21:55,820 --> 00:21:59,560
Of course, if the captain were mathematically inclined
282
00:21:59,560 --> 00:22:03,560
he'd try to figure out precisely
how to help the Coast Guard
283
00:22:03,560 --> 00:22:06,920
He'd determine his vessel's position
284
00:22:06,920 --> 00:22:10,180
perhaps, 150km west
285
00:22:10,180 --> 00:22:15,180
and 124km south of the Alameda Coast Guard base
286
00:22:15,180 --> 00:22:18,480
And of course, he'd come up with an estimate
of wind velocity
287
00:22:18,480 --> 00:22:21,840
say, 40km/h out of the south
288
00:22:21,840 --> 00:22:24,120
and the he'd calculate that his vessel
289
00:22:24,120 --> 00:22:30,020
is being pushed along at an average speed
of 2km/h due north
290
00:22:30,020 --> 00:22:34,240
However, as the men of the Irish Coffee
have come to realize
291
00:22:34,240 --> 00:22:37,140
Captain Duke, while he is called many things
292
00:22:37,140 --> 00:22:39,160
cannot be called a mathematician
293
00:22:39,480 --> 00:22:41,220
That's no criticism
294
00:22:41,220 --> 00:22:43,860
But when it comes to vectors, the coordinates
295
00:22:43,860 --> 00:22:46,720
to whatever may be used for a rescue
296
00:22:46,720 --> 00:22:49,280
it just doesn't have what it takes
297
00:22:49,280 --> 00:22:53,240
Unless that is, it takes a vivid imagination
298
00:22:54,260 --> 00:22:55,980
The way Captain Duke sees it
299
00:22:55,980 --> 00:22:59,420
the US Coast Guard wouldn't get very far without him
300
00:22:59,420 --> 00:23:02,160
So with his help, to say nothing of coordinates
301
00:23:02,160 --> 00:23:04,620
wind velocity and probable drift
302
00:23:04,620 --> 00:23:07,460
and a helicopter thrown in for good measure
303
00:23:07,460 --> 00:23:10,020
things can really take off
304
00:23:15,140 --> 00:23:17,440
In Captain Duke's imagination
305
00:23:17,440 --> 00:23:23,060
the pilot has enough fuel and
maximum aerial speed at 125km/h
306
00:23:23,060 --> 00:23:26,180
to remain in the air for hours
307
00:23:26,180 --> 00:23:30,060
That means 2 hours out, 2 hours back
308
00:23:30,060 --> 00:23:34,040
and 250km each way, no more
309
00:23:34,040 --> 00:23:37,300
Can the helicopter reach the Irish Coffee?
310
00:23:37,300 --> 00:23:39,140
and surely important
311
00:23:39,140 --> 00:23:42,040
Can the pilot make it back?
312
00:23:42,040 --> 00:23:46,280
Wind carries the copter due north, at 40km/h
313
00:23:46,280 --> 00:23:50,200
80km due north in 2 hours
314
00:23:50,200 --> 00:23:53,860
So, if the pilot were to fly along one of these vectors
315
00:23:53,860 --> 00:23:56,500
he'd wind up just out of range
316
00:23:56,500 --> 00:23:58,780
And even with Captain Duke's courage
317
00:23:58,780 --> 00:24:00,340
he'd fly no farther
318
00:24:00,340 --> 00:24:02,960
because he couldn't count on the steady wind to help him
319
00:24:02,960 --> 00:24:04,960
on his way back to base
320
00:24:06,260 --> 00:24:10,820
Still, the situation isn't as bad as it looks
in Captain Duke's worst scenario
321
00:24:10,820 --> 00:24:11,980
Why?
322
00:24:11,980 --> 00:24:15,220
Because there, the Irish Coffee drifts as well
323
00:24:15,220 --> 00:24:17,980
Heading north at 2km/h
324
00:24:18,840 --> 00:24:23,040
And in 2 hours, she'll drift 4km due north
325
00:24:23,040 --> 00:24:27,600
which would place her at the edge
of the chauffeur's range
326
00:24:27,600 --> 00:24:29,640
And accounting for aerial speed
327
00:24:29,640 --> 00:24:32,820
wind speed and drift velocity
328
00:24:32,820 --> 00:24:34,940
the pilot could calculate the right heading
329
00:24:34,940 --> 00:24:38,000
and reach the Irish Coffee
330
00:24:38,000 --> 00:24:40,000
Even in the Captain's fantasy
331
00:24:40,000 --> 00:24:42,600
is that possible?
332
00:24:42,600 --> 00:24:44,800
Yes. And in reality,
333
00:24:44,800 --> 00:24:47,300
the answer should be obvious by now
334
00:24:47,300 --> 00:24:51,080
The copter's path, and its displacement by the wind
335
00:24:51,080 --> 00:24:54,620
combine or add as vectors
336
00:24:54,620 --> 00:24:56,320
As simple as they are
337
00:24:56,320 --> 00:24:59,600
such calculations would be vital if the US Coast Guard
338
00:24:59,600 --> 00:25:02,000
read Captain Duke's mind
339
00:25:02,000 --> 00:25:05,780
Instead, they went along with a plan of their own
340
00:25:10,880 --> 00:25:14,200
We have a vessel to save southwest of the Fillmore...
341
00:25:14,200 --> 00:25:18,380
Actually, the Irish Coffee is a few km
south of the Farallon Islands
342
00:25:18,380 --> 00:25:20,280
near the San Francisco Bay
343
00:25:20,280 --> 00:25:22,980
And almost within shouting distance
344
00:25:23,700 --> 00:25:28,340
So soon enough, captain and crew will be, in good hands
345
00:25:28,340 --> 00:25:31,140
And long before their galley runs out of refreshments
346
00:25:31,140 --> 00:25:33,380
they'll be back on dry land
347
00:25:33,380 --> 00:25:36,640
And until they learn the lessons on seafaring
348
00:25:36,640 --> 00:25:38,640
to say nothing of vectors
349
00:25:38,640 --> 00:25:41,660
that's exactly where they belong
350
00:25:43,460 --> 00:25:46,040
Before Copernicus, the center of the Earth
351
00:25:46,040 --> 00:25:48,500
was the center of the universe
352
00:25:48,500 --> 00:25:50,420
At the Aristotelian world
353
00:25:50,420 --> 00:25:52,900
the very idea of place had no meaning
354
00:25:52,900 --> 00:25:54,480
except where something was
355
00:25:54,480 --> 00:25:56,700
with respect to the center of the Earth
356
00:25:58,260 --> 00:26:00,880
Then Copernicus comes along
357
00:26:00,880 --> 00:26:04,140
and thus affects the routine mathematical operation
358
00:26:04,140 --> 00:26:07,020
Writing the equations of the orbits of the planets
359
00:26:07,020 --> 00:26:09,020
in a different coordinate system
360
00:26:09,020 --> 00:26:11,260
And the world is never the same again
361
00:26:12,360 --> 00:26:14,040
But that's misleading
362
00:26:14,040 --> 00:26:16,040
It makes it sounds as if the important thing
363
00:26:16,040 --> 00:26:18,580
is to have the right coordinate system
364
00:26:18,580 --> 00:26:20,260
And what we learn from all that
365
00:26:20,260 --> 00:26:22,600
was really exactly the opposite
366
00:26:22,600 --> 00:26:26,440
That all coordinate systems are equally good
367
00:26:26,440 --> 00:26:29,800
Copernicus says the origin should be at the Sun
368
00:26:29,800 --> 00:26:33,680
the Coast Guard says the origin should be
at its coordinating station
369
00:26:33,680 --> 00:26:36,780
And they are both equally right
370
00:26:36,780 --> 00:26:40,000
In fact, we can say it in a way that's much more profound
371
00:26:40,000 --> 00:26:42,620
The idea is that the laws of physics
372
00:26:42,620 --> 00:26:44,980
are exactly the same everywhere
373
00:26:44,980 --> 00:26:48,160
Newton's laws work as well in the Crab Nebula
374
00:26:48,160 --> 00:26:50,960
as they do in Kansas City
375
00:26:50,960 --> 00:26:52,760
And because we believe that
376
00:26:52,760 --> 00:26:55,360
we need a way of expressing those laws
377
00:26:55,360 --> 00:26:59,380
that works equally well in all coordinate systems
378
00:26:59,380 --> 00:27:03,200
And that way, is by means, of vectors
379
00:27:04,240 --> 00:27:06,840
The idea of a vector is disconcerting
380
00:27:06,840 --> 00:27:09,760
because a vector has a size and a direction
381
00:27:09,760 --> 00:27:11,860
but it has no place
382
00:27:11,860 --> 00:27:13,980
But that's exactly what we need
383
00:27:13,980 --> 00:27:16,840
in order to express laws that are the same
384
00:27:16,840 --> 00:27:18,880
in every place
385
00:27:18,880 --> 00:27:21,000
In fact, next time
386
00:27:21,000 --> 00:27:23,020
we'll see a vector equation
387
00:27:23,020 --> 00:27:27,220
which lies at the heart of our understanding of the world
388
00:27:27,220 --> 00:27:29,220
I'll see you then
389
00:27:34,220 --> 00:28:05,000
Subtitle created by Tran Nguyen Phuong Thanh - 2014