"Annenberg Media" "¶" "CERTAINPHYSICALQUANTITIES HAVENOTONLYMAGNITUDE" "BUTDIRECTIONAS WELL." "THOSEQUANTITIES ARE REPRESENTED BY VECTORS." "SOME THINGS ARE STRICTLY A QUESTION" "OF YOUR POINT OF VIEW." "C0PERNICUS SAID EVERYONE HAS ALWAYS LOOKED AT THE UNIVERSE" "FROM THE POINT OF VIEW OF THE EARTH." "WHAT WOULD HAPPEN IF WE USED" "THE POINT OF VIEW OF THE SUN?" "YOUR POINT OF VIEW SHOULD NOT BE TAKEN LIGHTLY." "ONCE YOU'VE CHOSEN A POINT OF VIEW," "YOU NEED A MEANS OF DESCRIBING" "EXACTLY WHERE SOMETHING IS AND WHICH WAY IT'S GOING." "CHESS PLAYERS HAVE A SCHEME OF TELLING YOU" "WHERE EACH PIECE IS AND HOW IT MOVES." "CHILDREN PLAYING THE GAME OF BATTLESHIP" "UNDERSTAND HOW TO DESCRIBE WHERE SOMETHING IS." "ALONG WITH CHESS PLAYERS AND CHILDREN," "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" "IN THE SAN FRANCISCO BAY." "HERE, THE METHODS TO LOCATE OR LAUNCH A VESSEL" "ARE STRICTLY PROFESSIONAL." "FROM UNEVENTFUL HOURS" "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," "ARE THE TOOLS OF VECTOR MATHEMATICS." "AQUANTITYTHATHAS  BOTH MAGNITUDE AND DIRECTION" "ISA VECTOR, REPRESENTEDBYAN ARROW." "THEARROW'SDIRECTIONIS THEDIRECTIONOF THEQUANTITY," "ANDTHELENGTHOFTHEARROW  INDICATESITSMAGNITUDE." "DISPLACEMENT,VELOCITY, ANDACCELERATION" "AREALLVECTORS." "INEQUATIONS," "VECTORSAREWRITTEN INBOLDFACE." "ORDINARYQUANTITIESSUCHAS TIMEANDMASSARE SCALARS." "INEQUATIONS, SCALARS ARE WRITTEN IN ITALICS." "THEMAGNITUDEOF AVECTOR ISALSOASCALAR" "THESAMELETTER, BUTINITALICS." "WITHVECTORS," "EVENTHEMOSTFAMILIAR MATHEMATICALOPERATIONS" "ACQUIREANENTIRELY NEWMEANING." "FOR EXAMPLE, WHEN TWO VECTORS AREADDEDTOGETHER," "THESUMISN'TMERELYANUMBER." "IT'SA NEWVECTOR." "ALSO, VECTORS CAN BE SUBTRACTED FROMEACHOTHER," "ANDTHERESULTIS ANOTHERNEWVECTOR." "OF COURSE, WHETHER OR NOT" "THE COAST GUARD USES CLASSICAL SCIENTIFIC METHODS," "THEIR TOOLS ARE UP-TO-THE-MINUTE." "IRISHCOFFEE," "THIS IS SAN FRANCISCO VESSEL TRAFFIC." "AN OFFICER RECEIVES A DISTRESS SIGNAL." "HE'S PREPARED FOR WHATEVER THE FUTURE HOLDS" "AT THE OTHER END." "IN THE COURSE OF DUTY," "THE COAST GUARD RESCUES ALL MANNER OF SEA CREATURE." "THE COAST GUARD EVEN RESCUES" "THOSE WHO SEEM BETTER OFF ON DRY LAND." "A CREW OF SUNDAY SAILORS ON THE GOOD YACHT IRISH COFFEE." "THEY HAVE DRIFTED OFF COURSE" "WAY OFF COURSE, TO 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." "AT THE FLICK OF A SWITCH..." "[BEEPING]" "BOATCREWECHO, MANYOURBOAT." "THEY CAN SWING INTO ACTION." "SEAMANPOOLE, MANTHEREDDECK." "[HORN BLOWING]" "[HONK HONK]" "[HONK]" "BUT WAIT A SECOND." "THIS APPROACH MAY NOT BE NECESSARY." "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 TAKE THEIR COURAGE" "FROM THEIR LEADER, CAPTAIN DUKE." "WHETHER THE DANGER IS IMMEDIATE OR POTENTIAL," "IT'S NEVER TAKEN LIGHTLY BY THE COAST GUARD." "IRISH COFFEE," "THIS IS COAST GUARD." "LOCATION AND NATURE OF YOUR DISTRESS." "WE'RE IN TROUBLE." "WE'RE SOMEPLACE OFF THE COAST" "OF SAN FRANCISCO." "IRISH COFFEE," "REQUEST YOU MONITOR CHANNEL 13" "AND CONTACT US EVERY 15 MINUTES." "NOT ONLY ARE WE NOT SAILORS," "WE'RE NOT MAP READERS, EITHER." "IT'S GOT TO BE IN THE BLUE PART." "BEING CARELESSLY 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 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." "PICTURE SOME POINT ON THE MAP MARKED WITH THE RED DOT." "SUPPOSE THE RED DOT HAS" "THE HORIZONTAL COORDINATE, X, OF -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." "EACH MAP HAS ITS OWN SYSTEM OF COORDINATES." "BUT EVEN WITH FAMOUS LANDMARKS SUCH AS CENTRAL PARK..." "BOSTON COMMON," "OR THE WHITE HOUSE ON PENNSYLVANIA AVENUE," "MAPS CAN BEWILDER ONE" "WHO'S UNFAMILIAR WITH RECTANGULAR GRIDS." "OF COURSE, TO THE MAIL CARRIER," "DELIVERY PERSON, OR TAXICAB DRIVER," "THAT 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" "MET BY MEN WHO ACTUALLY UNDERSTOOD THE MEANING" "OF THE WORD "COORDINATES."" "IN THE HISTORY OF MATHEMATICS," "RENE DESCARTES AND PIERRE DE FERMAT" "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, ACROSS THE ENGLISH CHANNEL," "JOHN WALLIS DEVELOPED A THEORY ALONG SIMILAR LINES." "THOUGH KARL FRIEDRICH GAUSS LATER COINED 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." "NUMBERS ON A GRID MAP CAN LOOK COMPLEX." "IF THE PATHS OF THE CITIES SEEM PERPLEXING," "CONSIDER THE PLIGHT OF STRANGERS TO THE SEA." "LOOK, THERE'S THE PLANE." "GET THE LETTERS!" "HURRY UP BEFORE HE 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 SEVERAL DIFFERENT METHODS ON HAND," "INCLUDING TRIANGULATION." "TWO DIFFERENT ANGLES OF RADIO RECEPTION" "ARE ESTABLISHED BY USING DIRECTIONAL ANTENNAS." "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," "IF IT WERE STABLE ENOUGH TO STAY PUT." "THAT'S TOO MUCH TO EXPECT." "AT THE MERCY OF CURRENTS, VESSELS DRIFT ALL OVER THE SEA." "KNOWING HER LOCATION AT ONE FIXED TIME AND PLACE" "ISN'T ENOUGH." "THE COAST GUARD NEEDS THE DIRECTION AND THE SPEED" "OF IRISH COFFEE'S DRIFT." "WHICH WAY?" "HOW FAST?" "WHILE THE CAPTAIN HAS A NOVEL APPROACH," "HE WON'T FIND ANSWERS IN MOBY DICK." "FLYNN'S SMOKE SIGNALS ARE GOING NOWHERE," "SO ANOTHER DISTRESS SIGNAL" "PROBABLY CAN'T DO ANY MORE HARM." "WE'RE OFF THE COAST OF SAN FRANCISCO." "IF BOGART GAVE THE COAST GUARD A SECOND POSITION," "THE PICTURE COULD LOOK LIKE THIS." "BOTHPOSITIONS AREJOINEDBY AN ARROW" "FROMPOSITION1 TOPOSITION2." "THEARROW'SDIRECTION" "SHOWSWHICHWAY THEVESSELHASDRIFTED." "ITSLENGTHSHOWSHOW FAR ." "THISARROWIS CALLED ADISPLACEMENTVECTOR." "WITH MORE INFORMATION," "THE COAST GUARD CAN ESTIMATE" "THE PATH OF THE DRIFTING VESSEL" "BY JOINING" "THE DISPLACEMENT VECTORS" "TAIL TO HEAD," "WHICH IS VECTOR ADDITION." "THEY CAN ESTIMATE DRIFT VELOCITY" "BY DIVIDING THE VECTORS" "BY THE TIME BETWEEN SIGNALS." "A VECTOR CAN BE MULTIPLIED BYA SCALAR." "THISNEWVECTOR HASTHESAMEDIRECTION" "IFTHESCALARISPOSITIVE." "BUTITSMAGNITUDE CANBEDIFFERENT." "MULTIPLYINGA VECTOR BYA NEGATIVESCALAR" "REVERSESITSDIRECTION." "THE MODERN COAST GUARD" "RARELY ESTIMATES THE PATH OF DRIFTING VESSELS" "BY SKETCHING DISPLACEMENT AND VELOCITY VECTORS BY HAND." "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 NAMED HERMANN GRASSMANN." "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 HIS WORK AS TOO COMPLICATED" "AT THE TIME," "THE SEEDS BENEATH HERMANN GRASSMANN'S MATHEMATICS" "STEM FROM ANCIENT GREECE." "THE PARALLELOGRAM LAW OF COMPOSITION OF FORCES," "WHICH ARISTOTLE CONSIDERED IN THE SPECIAL CASE OF A RECTANGLE," "IS AN EXAMPLE OF ADDING VECTORS." "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." "EVEN THEN, IT WOULD TAKE A COUPLE OF HUNDRED YEARS" "AND WILLIAM HAMILTON" "TO SEE THE VECTOR'S SCOPE AND VERSATILITY." "ATTEMPTING TO MATHEMATICALLY INTERPRET ROTATION IN SPACE" "IN 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 THREE-DIMENSIONAL VECTOR PART" "AND A ONE-DIMENSIONAL SCALAR PART." "THOUGH IT WOULD TAKE SOME MODERN REFINEMENT" "IN THE VECTOR AND THE SCALAR," "HAMILTON HAD DEVISED A MODE OF EXPRESSING NEWTON'S MECHANICS." "JAMES CLERK MAXWELL," "A PIONEER IN ELECTRICITY AND MAGNETISM," "SAW THE THEORETICAL BRILLIANCE OF HAMILTON'S IDEAS." "EVEN SO, THEY WEREN'T GENERALLY ACCEPTED." "TO BE EFFECTIVE," "CONTENT AND FORM HAVE TO WORK TOGETHER." "WHILE HAMILTON'S THEORY WAS BRILLIANT," "ITS FORM WAS TOO COMPLICATED TO SURVIVE." "IN ESSENCE, HOWEVER, THE THEORY DID SURVIVE." "IN THE 19th CENTURY," "YALE PROFESSOR JOSIAH WILLIARD 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." "¶ ...99 BOTTLES OF BEER ¶" "¶ IF ONE OF THOSE BOTTLES SHOULD HAPPEN TO FALL... ¶" "OF COURSE, VECTOR ALGEBRA, LIKE MUSIC NOTATION," "TAKES SOME PRACTICE." "EVEN ABOARD THE IRISH COFFEE," "A LITTLE EXERCISE COULDN'T MAKE THE SITUATION MUCH WORSE." "IN ANY CASE, VECTORS, UNLIKE LYRICS OF CERTAIN SONGS," "ARE QUITE VERSATILE." "A VECTOR CAN BE MULTIPLIED BYANOTHERVECTOR." "THE DOT PRODUCT OF A AND B ISA SCALAR" "THAT MEASURES THE TENDENCY OFTHETWOVECTORS" "TO POINT IN THE SAME DIRECTION." "ADOTBISEQUALTO" "THELENGTHOF A TIMESTHELENGTHOFB" "TIMESTHECOSINE OFTHEANGLEBETWEENTHEM ." "IF A AND B ARE PERPENDICULAR, THEIR DOT PRODUCT IS ZERO." "THEDOTPRODUCT OFA VECTORWITHITSELF" "IS THE SQUARE OF ITS LENGTH." "¶ 38 BOTTLES OF BEER ON THE WALL ¶" "¶ 38 BOTTLES OF BEER... ¶" "MOVING RIGHT ALONG... ¶ SHOULD HAPPEN TO FALL... ¶" "THECROSSPRODUCT OFTWOVECTORS" "ISA NEWVECTOR" "PERPENDICULARTOTHEPLANE" "OF THE TWO ORIGINAL VECTORS." "ITSLENGTHIS THEAREA" "OFTHEPARALLELOGRAMFORMED" "BYTHEORIGINALVECTORS." "ITSDIRECTIONIS DETERMINED BYTHERIGHT-HANDRULE." "THEx PRODUCTMEASURES THE TENDENCY OF TWO VECTORS" "TOBEPERPENDICULAR," "ANDIT'SONEOFTHEMORE EFFECTIVETOOLS" "FORDESCRIBINGSPINNING ORROTATINGOBJECTS." "¶ 36 BOTTLES OF BEER ON THE WALL ¶" "¶ 35 BOTTLES OF BEER ON THE WALL ¶" "¶ IF ONE OF THOSE BOTTLES ¶" "¶ SHOULD HAPPEN TO FALL ¶" "¶ 34 BOTTLES OF BEER ON THE WALL... ¶" "VECTORALGEBRAMAKESUSE OF" "TWO PERPENDICULAR UNIT VECTORS, iANDj." "THELITTLEHATOVERTHEVECTOR" "MEANSTHATIT HASLENGTH1." "THEVECTORFROMTHE ORIGIN" "TOTHEPOINT" "WITHCOORDINATESx, y" "ISTHESUM" "OFTWOPERPENDICULARVECTORS" "AHORIZONTALVECTOR,xi," "ANDA VERTICALVECTOR,yj." "WITHTHEHELPOFi ANDj ," "ADDINGANDMULTIPLYINGVECTORS" "CANBEACCOMPLISHED" "BYORDINARYALGEBRA." "INTHREE-DIMENSIONALSPACE," "ATHIRDUNITVECTOR,k, ISUSED." "kISPERPENDICULARTO THEPLANEOF iAND j." "ATHREE-DIMENSIONALVECTOR" "ISWRITTENxi +yj+ zk ." "¶ 5 BOTTLES OF BEER ON THE WALL ¶" "¶ 5 BOTTLES OF BEER... ¶" "A MOVING SONG IS ONE THING." "A 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 GOOD, CLEAN FUN." "BUT CONDITIONS CHANGE," "AND THE IRISH COFFEE ISN'T WHERE OR WHAT SHE USED TO BE." "THE WIND AT SEA IS A FORCE TO BE RECKONED WITH." "BECAUSE WIND VELOCITY HAS SPEED AND DIRECTION," "IT'S A VECTOR QUANTITY." "SO IS THE VELOCITY OF FLOWING WATER." "DRIFTING WITH THE WIND AND THE WATER" "BUT GETTING NOWHERE FAST," "THESE MEN ARE AT A LOSS." "¶ 1 BOTTLE OF BEER ON THE WALL ¶" "¶ 1 BOTTLE OF BEER... ¶" "AND FINALLY, THEY REALIZE IT." "THAT'S IT." "IF THE CAPTAIN WERE MATHEMATICALLY INCLINED," "HE'D TRY TO HELP THE COAST GUARD." "HE'D DETERMINE" "HIS VESSEL'S POSITION" "PERHAPS 150 KILOMETERS WEST" "AND 124 KILOMETERS SOUTH" "OF THE ALAMEDA BASE." "HE'D ESTIMATE WIND VELOCITY," "SAY 40 KILOMETERS PER HOUR" "OUT OF THE SOUTH." "HE'D CALCULATE THAT HIS VESSEL" "IS BEING PUSHED ALONG" "AT AN AVERAGE SPEED" "OF 2 KILOMETERS PER HOUR" "DUE NORTH." "HOWEVER, AS THE MEN HAVE COME TO REALIZE," "CAPTAIN DUKE," "WHO IS CALLED MANY THINGS," "CANNOT BE CALLED A MATHEMATICIAN." "WHEN IT COMES TO VECTORS, TO COORDINATES," "TO WHATEVER MAY BE USED FOR A RESCUE," "HE JUST DOESN'T HAVE WHAT IT TAKES" "UNLESS IT TAKES A VIVID IMAGINATION." "CAPTAIN DUKE FIGURES THE U.S. COAST GUARD NEEDS HIM." "SO WITH HIS HELP" "TO SAY NOTHING OF COORDINATES, WIND VELOCITY," "PROBABLE DRIFT," "AND A HELICOPTER FOR GOOD MEASURE" "THINGS CAN REALLY TAKE OFF." "IN CAPTAIN DUKE'S IMAGINATION," "THE PILOT HAS ENOUGH FUEL" "AT MAXIMUM AIR SPEED OF 125 KILOMETERS PER HOUR" "TO REMAIN AIRBORNE FOUR HOURS." "THAT'S ROUGHLY TWO HOURS OUT, TWO HOURS BACK," "AND 250 KILOMETERS EACH WAY." "CAN THE HELICOPTER REACH THE IRISH COFFEE?" "CAN THE PILOT MAKE IT BACK?" "WIND CARRIES THE COPTER DUE NORTH" "AT 40 KILOMETERS PER HOUR, 80 KILOMETERS DUE NORTH IN TWO HOURS." "IF THE PILOT WERE TO FLY ALONG ONE OF THESE VECTORS," "HE'D WIND UP JUST OUT OF RANGE." "HE'D FLY NO FARTHER" "BECAUSE HE COULDN'T COUNT ON A STEADY WIND TO HELP HIM" "ON HIS WAY BACK TO BASE." "THE SITUATION ISN'T AS BAD AS IT LOOKS." "WHY?" "BECAUSE THE IRISH COFFEE DRIFTS AS WELL," "NORTHWARD AT 2 KILOMETERS PER HOUR." "IN TWO HOURS, SHE'LL DRIFT FOUR KILOMETERS DUE NORTH," "WHICH WOULD PLACE HER AT THE EDGE OF THE CHOPPER'S RANGE." "ACCOUNTING FOR AIR SPEED, WIND SPEED, AND DRIFT VELOCITY," "THE PILOT WOULD 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." "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 U.S. COAST GUARD READ CAPTAIN DUKE'S MIND." "INSTEAD, THEY WENT ALONG WITH A PLAN OF THEIR OWN." "BOATCREWECHO," "WEHAVEAVESSELDISABLED" "SOUTHWESTOF THEFARALLONISLANDS." "THE IRISH COFFEE IS JUST SOUTH OF THE FARALLON ISLANDS," "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." "UNTIL THEY LEARN THEIR LESSONS ON SEAFARING," "THAT'S EXACTLY WHERE THEY BELONG." "BEFORE COPERNICUS, THE CENTER OF THE EARTH" "WAS THE CENTER OF THE UNIVERSE." "THE VERY IDEA OF PLACE HAD NO MEANING" "EXCEPT WHERE SOMETHING WAS" "WITH RESPECT TO THE CENTER OF THE EARTH." "COPERNICUS DOES, IN EFFECT," "A ROUTINE MATHEMATICAL OPERATION," "WRITING THE EQUATIONS OF THE PLANETARY ORBITS" "IN A DIFFERENT COORDINATE SYSTEM." "THE WORLD IS NEVER THE SAME AGAIN." "BUT THAT'S MISLEADING." "IT MAKES IT SOUND LIKE THE IMPORTANT THING IS TO HAVE" "THE RIGHT COORDINATE SYSTEM." "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." "WE CAN SAY IT MORE PROFOUNDLY." "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." "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." "THAT'S WHAT WE NEED TO EXPRESS LAWS" "THAT ARE THE SAME EVERYWHERE." "NEXT TIME, WE'LL SEE A VECTOR EQUATION" "WHICH LIES AT THE HEART OF OUR UNDERSTANDING" "OF THE WORLD." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION AND THE ANNENBERG/CPB PROJECT" "PUBLIC PERFORMANCE OF CAPTIONS PROHIBITED WITHOUT PERMISSION OF NATIONAL CAPTIONING INSTITUTE" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"