foreign media [Music] [Applause] [Music] [Applause] [Music] thank you thank you certain physical quantities have not only magnitude but direction as well those quantities are represented by vectors 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 looked at it from the point of view of the Sun and you know what the result of that was well 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's 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 and 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 crisis on the high seas the 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 U. S 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 [Music] the Arrow's direction is the direction of the quantity and the length of the arrow indicates its magnitude [Music] 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 [Music] 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 meeting 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 foreign of course whether or not the Coast Guard uses classical scientific methods their tools are up to the minute and ready to be called upon Irish coffee this is San Francisco vessel traffic cover an officer receives a signal of distress and no matter what the future may hold at the other end well prepared he has to be in the normal course of Duty the Coast Guard rescues all manner of sea creature and as a matter of 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 have drifted off course way off course two points entirely unknown likewise unknown is 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 [Music] they can swing into action [Music] um but wait a second this approach may not be necessary [Music] 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 chip Irish coffee Malcolm Bogart and Percival Flynn Brave man who on the high seas take their courage from their leader Captain Duke however no matter whether the danger is immediate or merely potential it's never taken lightly by the United States Coast Guard Irish coffee this is the San Francisco Coast Guard location and they trigger distress over this is the Irish coffee repeat we're in trouble and we're someplace off the coast of San Francisco Irish coffee the San Francisco Coast Guard equestrian monitor Channel 13 and contact us every 15 minutes over [Music] being carelessly lost at sea is no laughing matter and an uneducated guess can make this situation perilous at best particularly when these coordinates are on the San Francisco road map nonetheless there's something to be learned here these lines form a rectangular Grid at least to the extent that 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 the Red Dot suppose the Red Dot has the horizontal coordinate X of minus 150 kilometers and suppose it has a vertical coordinate y of -124 kilometers whatever the distances 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 all its own but even with world famous landmarks such as Central Park Boston Common or the White House on Pennsylvania Avenue Maps can be wilder one who's unfamiliar with rectangular grids of course to the mail carrier firefighter 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 men who actually understood the meaning of the word coordinates in the history of mathematics Rene Descartes and Pierre de Vermont wrote the opening chapter on coordinate systems in the 17th century these French Scholars cleverly plotted a scheme to connect geometry and algebra at the same time but across the English Channel John Wallace developed a theory along similar lines though Carl Friedrich Gauss would later coin the term Wallace was the first to introduce the idea of the complex number in his book algebra Wallace represented a complex number A Plus bi for example by measuring the real part a along a horizontal axis and the imaginary Part B along a vertical axis [Music] 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 look break up [Music] 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 copy would be just about here that is just about here if the Irish coffee were stable enough to stay put but that's too much to expect because at the mercy of currents vessels drift all over the sea clearly 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 coffees 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 we're off the coast of San Francisco please come in Coast Guard repeat 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 one to position two the Arrow's Direction shows which way the vessel is 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 Irishman and a German by the name of Hermann Grossman for his part in the creation of this mathematics Grassman tried to develop an algebraic structure on which geometry of any number of Dimensions could be based although many viewed his work as too complicated at the time the seeds beneath Hermann grassman's mathematics stem from deep within the intellect of ancient Greece after all the parallelogram law of composition of forces which Aristotle considered in the special case of a rectangle is 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 a mathematical way to interpret rotation in space and physical terms Hamilton sought a three-dimensional analog of complex numbers instead he found an algebra of four-dimensional objects 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 viable mode of expressing Newton's mechanics James Clark 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 at Yale Josiah Willard Gibbs took original ideas from Hamilton and Grossman 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 [Music] of course vector algebra like music notation takes some practice and even aboard the Irish coffee a little exercise couldn't make the 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 of the two vectors to point in the same direction a dot 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 [Music] [Applause] moving right along [Music] 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 the original vectors [Music] its direction is determined by the right hand rule 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 34 [Music] .
vector algebra makes use of two perpendicular unit vectors I and J the little hat over the vector means that it has length one the vector from the origin to the point with coordinates X Y is the sum of two perpendicular vectors A horizontal Vector x i and a vertical Vector YJ [Music] with the help of I and J adding and multiplying vectors can be accomplished by ordinary algebra [Music] foreign [Music] space a third unit vector k is used K is perpendicular to the plane of I and J three-dimensional Vector is written x i plus y j [Music] plus z k [Music] five bottles of beer on the wall five bottles of beer and if one of those models 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 clean fun 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 but getting nowhere fast these men are at a loss and finally they realize that's it of course if the captain were mathematically inclined he'd tried to figure out precisely how to help the Coast Guard he determined his vessel's position [Music] perhaps 150 kilometers West and 124 kilometers south of the Alameda Coast Guard base and of course he'd come up with an estimate of wind velocity say 40 kilometers per hour out of the South and then he'd calculate that his vessel is being pushed along at an average speed of two kilometers per hour 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 to coordinates to whatever may be used for a rescue he just doesn't have what it takes unless that is it takes a vivid imagination the way Captain Duke sees it the U. S 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 a pilot has enough fuel at maximum airspeed of 125 kilometers per hour to remain in the air four hours that's roughly two hours out two hours back and 250 kilometers each way no more can the helicopter reach the Irish coffee and surely is important can the pilot make it back wind carries the copter due north at 40 kilometers per hour 80 kilometers due north in two hours so if the pilot were to fly along one of these vectors you'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 a 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 frame of mind why because there the Irish coffee drifts as well northward at two kilometers per hour and in two hours she'll drift four kilometers due north which would Place her at the edge of the Chopper's range and accounting for AirSpeed wind speed and drift velocity the pilot would calculate the right heading and reach the Irish coffee [Music] 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 that the U. S Coast Guard red Captain Duke's mind instead they went along with the plan of Their Own [Music] actually the Irish coffee is a few kilometers south of the Farallon islands near the San Francisco Bay and almost within shouting distance 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 their 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 in fact in 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 doesn't affect a 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 sound as if the important thing is to have the right coordinate system and what we learned from all that was really exactly the opposite that all coordinate systems are equally good Copernicus says the origin should be in the Sun the Coast Guard says the origin should be in its coordinating station and they're 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 in 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 [Music] Annenberg media for information about this and other Annenberg media programs call 1-800 learner and visit us at www.
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