"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 silver 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"