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[ "A rotation of the orthonormal basis in terms of which vector a is represented is obtained with a multiplication of a by a rotation matrix R. This matrix multiplication is just a compact representation of a sequence of dot products .", "A rotation of the orthonormal basis in terms of which vector is represented i...
[ "The dot product is worked out by multiplying and summing across a single index in both tensors .", "The dot product is calculated by multiplying and summing across a single index in both tensors ." ]
[ "This is called the scalar projection of a onto b , or scalar component of a in the direction of b ( see figure ) .", "This is called the scalar projection of onto , or scalar component of in the direction of ( see figure ) ." ]
[ "Two non-zero vectors a and b are perpendicular if and only if a â cents b = 0 .", "Two non-zero vectors a and b are orthogonal if and only if a â cents b = 0 ." ]
[ "For a vector a , the dot product a  a is the square of the length of a , or", "For a vector , the dot product is the square of the length of , or" ]
[ "In mathematics , the dot product is an algebraic operation that takes two equal-length sequences of numbers ( usually coordinate vectors ) and returns a single number obtained by multiplying corresponding entries and adding up those products .", "In mathematics , the dot product is an algebraic operation that ta...
[ "Each of these dot products determines a scalar component of a in the direction of a rotated basis vector ( see previous section ) .", "Each of these dot products determines a scalar component of in the direction of a rotated basis vector ( see previous section ) ." ]
[ "Moreover , two vectors can be considered orthogonal if and only if their dot product is zero , and they both have a nonzero length .", "Moreover , two vectors can be considered orthogonal if and only if their dot product is zero , and they have non-null length ." ]
[ "If both a and b have length one ( i.e. , they are unit vectors ) , their dot product simply gives the cosine of the angle between them .", "If both and have length one ( i.e. , they are unit vectors ) , their dot product simply gives the cosine of the angle between them ." ]
[ "If only b is a unit vector , then the dot product a  b gives a cos ( Π) , i.e. , the magnitude of the projection of a in the direction of b , with a minus sign if the direction is opposite .", "If only is a unit vector , then the dot product gives , i.e. , the magnitude of the projection of in the direction of...
[ "where a denotes the length ( magnitude ) of a. More generally , if b is another vector", "where ( also written as ) denotes the length ( magnitude ) of . More generally , if is another vector" ]
[ "As the cosine of 90 Â is zero , the dot product of two orthogonal ( perpendicular ) vectors is always zero .", "As the cosine of 90 Â is zero , the dot product of two orthogonal vectors is always zero ." ]
[ "In physics , magnitude is a scalar in the physical sense , i.e. a physical quantity independent of the coordinate system , expressed as the product of a numerical value and a physical unit , not just a number .", "In physics , vector magnitude is a scalar in the physical sense , i.e. a physical quantity independ...
[ "where Î # denotes summation notation ( the sum of all the terms ) and n is the dimension of the vector space .", "where Î # denotes summation notation and n is the dimension of the vector space ." ]
[ "By definition , Ra1 consists of a sequence of dot products between each of the three rows of R and vector a1 .", "By definition , consists of a sequence of dot products between each of the three rows of and vector ." ]
[ "The same way , in a dimension 3 , the dot product of vectors ( a , b , c ) and ( d , e , f ) is ad + be + cf . For example , the dot product of two three-dimensional vectors ( 1 , 3 , â ' 5 ) and ( 4 , â ' 2 , â ' 1 ) is", "Similarly , in a dimension 3 , the dot product of vectors ( a , b , c ) and ( d , e , f )...
[ "where a and b denote the length of a and b and Î is the angle between them .", "where and denote the length of and and is the angle between them ." ]
[ "Lovendegem is a municipality located in the Belgian province of East Flanders .", "Lovendegem is a municipality located in Flanders , one of the three regions of Belgium , and in the province of East Flanders ." ]
[ "She starred as Carol Brady in The Brady Bunch between 1969 and 1974 .", "Florence Agnes Henderson ( born February 14 , 1934 ) is an American actress and singer , best known for playing the role of Carol Brady in the television program The Brady Bunch , which ran from 1969 to 1974 ." ]
[ "She is currently hosting her own talk show , The Florence Henderson Show , on television .", "As of 2010 , she hosts the American TV talk show The Florence Henderson Show ." ]
[ "The Italian Wikipedia ( In Italian : Wikipedia in italiano ) is the Italian-language edition of Wikipedia .", "The Italian Wikipedia is the Italian-language edition of Wikipedia ." ]
[ "This edition was started in January , 2002 and has over 783,831 articles .", "This edition was started in January 2002 and has over articles and more than registered accounts ." ]
[ "Paul Stanley is a solo album from the guitarist of the American hard rock\\/heavy metal band Kiss .", "Paul Stanley is a 1978 album from the rhythm guitarist and lead vocalist of American hard rock band Kiss ." ]
[ "It was released on September 18 , 1978 , as one of four solo albums released by the members of Kiss .", "It was one of four solo albums released by the members of Kiss on September 18 , 1978 ." ]
[ "Tux Racer is a video game made by Sunspire Studios where the player moves Tux , the Linux Penguin .", "Tux Racer is a Free Software 3D computer game starring the Linux mascot , Tux the penguin ." ]
[ "There are many things that slow Tux down .", "Sliding on ice makes Tux go faster , while sliding on snow allows for more maneuverability and sliding on rocky patches will slow Tux down . There are also trees to block Tux 's path and flags for sake of marking out the course ." ]
[ "Brandon is a city in Manitoba , Canada . It is the second largest city in Manitoba .", "Brandon is the second largest city in Manitoba , Canada , and is located in the southwestern area of the province ." ]
[ "Autreville is a commune . It is found in the region Picardie in the Aisne department in the north of France .", "Autreville is a commune in the department of Aisne in Picardy in northern France ." ]
[ "Doak led the National League in ERA again in 1921 .", "Doak led the NL in ERA again in 1921 ." ]
[ "Bill Doak ( born January 28 , 1891 , in Pittsburgh , Pennsylvania ) played 11 years with the St. Louis Cardinals .", "William Leopold Doak was born January 28 , 1891 , in Pittsburgh , Pennsylvania . He played 11 years with the St. Louis Cardinals teams ." ]
[ "In , he went 20-6 with a league leading 1.72 ERA .", "In , he went 19-6 with a league leading 1.72 ERA ." ]
[ "The Roman Catholic bishop of Sion is the oldest in Switzerland and one of the oldest north of the Alps .", "The Roman Catholic diocese of Sion is the oldest in Switzerland and one of the oldest north of the Alps ." ]
[ "This incinerator is equipped with a catalytic converter , something that reduces harmful gases to below unacceptable levels .", "This incinerator is equipped with a catalytic converter which lowers the levels of NOx emissions below acceptable levels ." ]
[ "The tertiary sector is the main economic sector in Sion .", "The tertiary sector is the main economic sector in Sion , mainly due to the presence of the cantons administration , the Valaisan parliament and the cantons courthouse ." ]
[ "Finally , the commune hosts a waste incinerator which disposes of the waste from 44 surrounding communes .", "Finally the commune hosts a waste incinerator which disposes of the waste from 44 surrounding communes ." ]
[ "The heat created from the burning is converted into electrical energy that is then provided to local communes .", "In addition , the heat created from the combustion is converted into electrical energy which is then provided to local communes ." ]
[ "Elk City is a city of Oklahoma in the United States .", "Elk City is a city in Beckham County , Oklahoma , United States ." ]
[ "Tottenham is an urban area of North London , England in the London Borough of Haringey . It is situated 6.6 m north-east of Charing Cross .", "Tottenham ( Ë tÉ ' Ê '' nÉ m ) is a district of north London , England in the London Borough of Haringey , situated 6.6 m north-northeast of Charing Cross ." ]
[ "OndÅ ej Sosenka holds the UCI hour record at 49.700 kilometres .", "The current UCI record is by OndÅ ej Sosenka , 49.700 km ( 30.882 mi ) ." ]
[ "Collingwood is a suburb in Melbourne , Australia . It was named after Baron Cuthbert Collingwood ( 1748 - 1810 ) , an admiral of the Royal Navy .", "It was named after Baron Collingwood , a British admiral ." ]
[ "The Collingwood Football Club was originally based in Collingwood , but has since moved to the Lexus Centre .", "The ornate Collingwood Town Hall is an imposing building that was used until 1994 as the center of local government for the City of Collingwood ." ]
[ "Bismarck Barreto Faria ( born 17 September 1969 ) is a former Brazilian football player .", "Bismarck Barreto Faria , best known as Bismarck ( born November 11 , 1969 in Sà # o Gonà alo , Rio de Janeiro State ) is a former Brazilian football ( soccer ) player ." ]
[ "A gazelle is a kind of antelope . Gazelles form the genera Gazella , Eudorcas and Nanger .", "The gazelle species are classified in the Gazella , Eudorcas and Nanger ." ]
[ "Gazelles are mostly found in the grasslands and savannas of Africa , but they are also found in southwest Asia .", "Gazelles are mostly found in the deserts , grasslands and savannas of Africa , but they are also found in southwest and central Asia , and the Indian subcontinent ." ]
[ "Gazelle comes from from the Arabic ghazÄ l .", "Gazelle is derived from the Arabic ghazÄ l ." ]
[ "It is an internationally well-known orchestra . Their conductor today is Yuri Temirkanov .", "Today it is an internationally recognized symphony orchestra under the directorship of Yuri Temirkanov ." ]
[ "By 1991 Leningrad had changed its name again to Saint Petersburg , so the orchestra was renamed Saint Petersburg Philharmonic Orchestra .", "In 1991 the orchestra gained its current name after its home city returned to its original name of Saint Petersburg ." ]
[ "Around this time the orchestra changed its name to Leningrad Philharmonic Orchestra , because the city had changed its name again , this time to `` Leningrad '' .", "Around this time the orchestra was renamed the Leningrad Philharmonic Orchestra ." ]
[ "By the 1900s it had started to give public performances .", "By the 1900s it had started to give public performances at the Philharmonia and elsewhere ." ]
[ "The Saint Petersburg Philharmonic Orchestra is an orchestra based in Saint Petersburg in Russia . It is the oldest orchestra in Russia .", "The Saint Petersburg Philharmonic Orchestra was formed in 1882 and is Russia 's oldest symphony orchestra ." ]
[ "After the Russian Revolution the country no longer had a tsar . The town of Saint Petersburg changed its name to Petrograd , so the orchestra 's name changed to `` State Philharmonic Orchestra of Petrograd '' .", "After the Russian Revolution the orchestra was taken over by its members , who changed the name to ...
[ "In the 1920s the orchestra began to get money from the state . It was an excellent orchestra and was famous abroad .", "In the 1920s the orchestra began receiving support from the state , and began to be known internationally as an excellent orchestra ." ]
[ "Guest conductors who conducted the orchestra included Bruno Walter , Ernest Ansermet and Hans Knappertsbusch at this time .", "Guest conductor appearances were made by Bruno Walter , Ernest Ansermet and Hans Knappertsbusch at this time ." ]
[ "Naomi Linnell was his partner in the last years of his life .", "Naomi Linnell was his partner for the last twenty three years of his life ." ]
[ "He died in Broadstairs , near his home on the Kent coast , on 8 December 2008 .", "Postgate died peacefully at a nursing home in Broadstairs , near his home on the Kent coast , on 8 December 2008 , aged 83 ." ]
[ "Oliver Postgate ( born Hendon , Middlesex , England , 12 April 1925 died Broadstairs , Kent , 8 December 2008 ) was an English animator , puppeteer and writer .", "Oliver Postgate ( 12 April 1925 â `` 8 December 2008 ) was an English animator , puppeteer and writer ." ]
[ "Postgate married Prudence Myers in 1957 , becoming stepfather to her three children The couple had twin boys in 1959 and another son in 1964 Prudence died in 1982 .", "Postgate married Prudence Myers in 1957 , becoming stepfather to her three children ( Kevan , Kerris and Krispian ) . The couple had twins in 195...
[ "In 1987 the University of Kent at Canterbury awarded an honorary degree to Postgate . Postgate said that it should really be Bagpuss who got the degree .", "In 1987 the University of Kent at Canterbury awarded an honorary degree to Postgate , who stated that the degree was really intended for Bagpuss , who was s...
[ "They include Pingwings , Pogles ' Wood , Noggin the Nog , Ivor the Engine , Clangers and Bagpuss . They were all made by Smallfilms , the company he set up with Peter Firmin .", "Pingwings , Pogles ' Wood , Noggin the Nog , Ivor the Engine , Clangers and Bagpuss , were all made by Smallfilms , the company he set...
[ "Lungern is a municipality of the canton of Obwalden in Switzerland .", "Lungern is a municipality in the canton of Obwalden in Switzerland ." ]
[ "Conway is a city of Iowa in the United States .", "Conway is a city in Taylor County , Iowa , United States ." ]
[ "Andrew `` Andy '' Murray ( born 15 May 1987 ) is a Scottish professional tennis player .", "Andrew `` Andy '' Murray ( born 15 May 1987 ) is a Scottish professional tennis player and current British No. 1 ." ]
[ "With help from Professor Frink , Homer is able to destroy the aliens ' spaceship , and Springfield and its people are returned to normal .", "With help from Professor Frink , however , Homer is able to destroy the aliens ' spaceship , and Springfield and its inhabitants are returned to normal , for Springfield s...
[ "The Simpsons Hit & Run is an action-adventure video game based on the animated sitcom ( comedy TV show ) The Simpsons .", "The Simpsons Hit & Run is an action-adventure video game based on the animated sitcom The Simpsons ." ]
[ "The game received mostly positive , or good reviews from video game critics .", "The game received generally favorable reviews from video game critics ." ]
[ "More than three million copies of the game were sold as of June 2007 .", "As of June 2007 , over three million copies of the game were sold ." ]
[ "When the Simpsons themselves decide to find out what is going on , they find out that aliens Kang and Kodos are filming a reality television series about the city .", "When several of the citizens take matters into their own hands , they discover that two aliens Kang and Kodos are filming a reality television se...
[ "he game 's playable characters are Homer Simpson , Bart Simpson , Lisa Simpson , Marge Simpson , and Apu Nahasapeemapetilon .", "The game 's playable characters are Homer Simpson , Bart Simpson , Lisa Simpson , Marge Simpson , and Apu Nahasapeemapetilon ." ]
[ "The game follows the Simpson family and the people who live in Springfield , who see many strange things that happen in their town .", "The game follows the Simpson family and the citizens of Springfield , who witness many strange incidents that occur in Springfield ." ]
[ "The player can only move , or control one certain character in each level .", "The player can only control one specific character in each level ." ]
[ "The voices were supplied by The Simpsons TV show actual cast .", "The story and dialogue were crafted by writers from The Simpsons , with all character voices supplied by the actual cast ." ]
[ "Praise was given mostly on how much the game is was like the show and its funny take on the Grand Theft Auto III video game .", "Praise focused on the interpretation of the Simpsons television series as a video game and its parodical take on the Grand Theft Auto III video game , while criticism mostly surrounded...
[ "When walking , the playing character can walk , run , and perform many attacks : a normal kick , a jumping kick , and a smashing move .", "When traveling by foot , the player character can walk , run , and perform several attacks : a normal kick , a jumping kick , and a smashing move ." ]
[ "The game 's driving missions are also similar to those of Grand Theft Auto 3 .", "The game 's driving missions are also similar to those of Grand Theft Auto III ." ]
[ "The game was developed or made by Radical Entertainment and published or was put out to be sold by Vivendi Universal .", "The game was developed by Radical Entertainment and published by Vivendi Universal Games ." ]
[ "In both games , the player races against other characters , gets items before a timer runs out , and wrecks other cars .", "In both games , the player races against other characters , collects items before a timer runs out , and wrecks other cars ." ]
[ "To drive , the player can either drive one of the many vehicles that are on the road , a feature taken from Grand Theft Auto 3 , or use a phone booth to choose a car .", "To drive , the player can either commandeer one of the vehicles that litter the road , a feature taken from Grand Theft Auto III , or use a ph...
[ "To make the show more interesting , the aliens release Buzz Cola , a soda that makes people go crazy , into Springfield 's water supply .", "To make the show more interesting , the aliens release a new version of the popular soft drink Buzz Cola that makes people go insane , into Springfield 's water supply ." ]
[ "It was released for the GameCube , Xbox , PlayStation 2 , and Windows in the United States on September 16 , 2003 and in Europe on October 31 , 2003 .", "It was released for the GameCube , Xbox , PlayStation 2 , and Windows in North America on September 16 , 2003 , In Europe and Australia on October 31 , 2003 an...
[ "Montigny-Lengrain is a commune . It is found in the region Picardie in the Aisne department in the north of France .", "Montigny-Lengrain is a commune in the Aisne department in Picardy in northern France ." ]
[ "A centimetre ( the American spelling is centimeter ; symbol is cm ) is a unit of length in the SI measurement system .", "A centimetre ( American spelling : centimeter , symbol cm ) is a unit of length in the metric system , equal to one hundredth of a metre , which is the current SI base unit of length ." ]
[ "The Lincoln Memorial Reflecting Pool is the biggest reflecting pool in Washington , D.C. .", "The Lincoln Memorial Reflecting Pool is the largest of the many reflecting pools in Washington , D.C. , U.S. ." ]
[ "It is right next to the Lincoln Memorial . The reflecting pool is long and can be seen in many photographs of the Washington Monument .", "It is a long and large rectangular pool located on the National Mall , directly east of the Lincoln Memorial , with the Washington Monument to the west ." ]
[ "There are paths and trees on both sides , and many people exercise around it .", "It is lined by walking paths and shade trees on both sides ." ]
[ "In 1962 , Crosby became the first person ever to receive the Grammy Lifetime Achievement Award .", "In 1962 , Crosby was given the Grammy Lifetime Achievement Award ." ]
[ "Pocahontas is a city of Iowa in the United States .", "Pocahontas is a city in Pocahontas County , Iowa , United States ." ]
[ "Rocks also ranked # 176 on Rolling Stones list of the 500 Greatest Albums of All Time and is also noted in musician Kurt Cobain 's top 50 favorite albums in his journals .", "Rocks also ranked # 176 on Rolling Stones list of the 500 Greatest Albums of All Time ." ]
[ "Also , it has greatly influenced several people in the hard rock and heavy metal community , including members of Guns N ' Roses and Metallica .", "Additionally , it has greatly influenced several people in the hard rock and heavy metal community , including members of Guns N ' Roses and Metallica ." ]
[ "It has five games on it - Half-Life 2 , Half-Life 2 : Episode One ( an expansion of Half-Life 2 ) , Half-Life 2 : Episode Two ( another expansion of Half-Life 2 , Portal , and Team Fortress 2 . The first two were released outside of The Orange Box , but the other three first appeared in it .", "The soundtrack fo...
[ "It has five games on it - Half-Life 2 , Half-Life 2 : Episode One ( an expansion of Half-Life 2 ) , Half-Life 2 : Episode Two ( another expansion of Half-Life 2 , Portal , and Team Fortress 2 .", "Two of the games included , Half-Life 2 and its first stand-alone expansion , Episode One , had previously been rele...
[ "Under the Hookers , the Royal Botanical Gardens at Kew rose to world renown .", "Under the directorship of father and son Hooker , the Royal Botanical gardens of Kew rose to world renown ." ]
[ "Sir Joseph Dalton Hooker OM GCSI CB MD FRS ( 30 June 1817 â `` 10 December 1911 ) was a British botanist and explorer of the 19th century .", "Sir Joseph Dalton Hooker OM , GCSI , CB , MD , FRS ( 30 June 1817 -- 10 December 1911 ) was one of the greatest British botanists and explorers of the 19th century ." ]
[ "They had met once , before the Antarctic voyage started .", "They had met once , before the Antarctic voyage embarked ." ]
[ "After spending 5 months in the Antarctic they returned to resupply in Hobart , then went on to Sydney , and New Zealand .", "After spending 5 months in the Antarctic they returned to resupply in Hobart , then went on to Sydney , and the Bay of Islands in New Zealand ." ]
[ "He sailed on the Erebus and was assistant to the ship 's surgeon , who was instructed to collect zoological and geological specimens .", "He sailed on the Erebus and was assistant to Robert McCormick , who in addition to being the ship 's Surgeon was instructed to collect zoological and geological specimens ." ]
[ "He received three of its medals : the Royal Medal in 1854 , the Copley Medal in 1887 and the Darwin Medal in 1892 .", "He received three of its medals : the Royal Medal in 1854 , the Copley in 1887 and the Darwin Medal in 1892 ." ]
[ "Hooker 's survey of hitherto unexplored regions , the Himalayan Journals was published in 1854 .", "Hooker 's survey of hitherto unexplored regions , the Himalayan Journals , dedicated to Charles Darwin , was published by the Calcutta Trigonometrical Survey Office in 1854 , abbreviated again in 1855 and later by...
[ "In 1855 he was appointed Assistant-Director of the Royal Botanic Gardens , Kew , and in 1865 he succeeded his father as Director , holding the post for twenty years .", "In 1855 he was appointed Assistant-Director of the Royal Botanic Gardens , Kew , and in 1865 he succeeded his father as full Director , holding...
[ "They traveled north west along Nepal 's passes into Tibet .", "They traveled to Zongri , west over the spurs of Kangchenjunga , and north west along Nepal 's passes into Tibet ." ]
[ "Hooker was one of the founders of geographical botany , and Charles Darwin 's closest friend .", "Hooker was a founder of geographical botany , and Charles Darwin 's closest friend ." ]