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106-474: The Seaplane Experimental Station , formerly RNAS Felixstowe , was a British aircraft design unit during the early part of the 20th century. During June 1912, surveys began for a suitable site for a base for Naval hydro-aeroplanes, with at first Shotley or Mistley on the River Stour . Ultimately Felixstowe was chosen and the formation of the new Naval Air Station, along with another at Great Yarmouth

212-426: A {\displaystyle {\mathfrak {a}}} . Vectors are usually shown in graphs or other diagrams as arrows (directed line segments ), as illustrated in the figure. Here, the point A is called the origin , tail , base , or initial point , and the point B is called the head , tip , endpoint , terminal point or final point . The length of the arrow is proportional to the vector's magnitude , while

318-473: A z = a x i + a y j + a z k . {\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.} The notation e i is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. As explained above ,

424-854: A 1 a 2 a 3 ] = [ a 1   a 2   a 3 ] T . {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=[a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }.} Another way to represent a vector in n -dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) . {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).} These have

530-412: A 1 , a 2 , a 3 ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).} also written, a = ( a x , a y , a z ) . {\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).} This can be generalised to n-dimensional Euclidean space (or R ). a = (

636-397: A 1 , a 2 , a 3 , ⋯ , a n − 1 , a n ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).} These numbers are often arranged into a column vector or row vector , particularly when dealing with matrices , as follows: a = [

742-598: A 3 ( 0 , 0 , 1 ) ,   {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ } or a = a 1 + a 2 + a 3 = a 1 e 1 + a 2 e 2 + a 3 e 3 , {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},} where

848-721: A 1 , a 2 , a 3 are called the vector components (or vector projections ) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x , y , and z (see figure), while a 1 , a 2 , a 3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often denoted i , j , k {\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } instead (or x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } , in which

954-412: A . ( Uppercase letters are typically used to represent matrices .) Other conventions include a → {\displaystyle {\vec {a}}} or a , especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, e.g. a ∼ {\displaystyle {\underset {^{\sim }}{a}}} , which

1060-418: A directed line segment , or arrow, in a Euclidean space . In pure mathematics , a vector is defined more generally as any element of a vector space . In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of

1166-600: A Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → , {\displaystyle {\overrightarrow {E}},} and a group action of the additive group of E → , {\displaystyle {\overrightarrow {E}},} which is free and transitive (See Affine space for details of this construction). The elements of E → {\displaystyle {\overrightarrow {E}}} are called translations . It has been proven that

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1272-442: A basis does not affect the properties of a vector or its behaviour under transformations. A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively normal , and tangent to a surface (see figure). Moreover, the radial and tangential components of

1378-728: A condition may be emphasized calling the result a bound vector . When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector . The distinction between bound and free vectors is especially relevant in mechanics, where a force applied to a body has a point of contact (see resultant force and couple ). Two arrows A B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{AB}}} and A ′ B ′ ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}} in space represent

1484-431: A connection to London with intermediate stops at other major towns, such as Ipswich, Colchester , Witham , Chelmsford and Brentwood . A weekday service departs daily at 07:20 and returns at 18:00, while the weekend service departs at 07:55 and returns at 18:00. Felixstowe is administered by Suffolk County Council , East Suffolk Council and Felixstowe Town Council. The local parish council, Felixstowe Town Council,

1590-414: A convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the cross product , which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of

1696-436: A convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector ( 1 , 2 , 3 ) + ( − 2 , 0 , 4 ) = ( 1 − 2 , 2 + 0 , 3 + 4 ) = ( − 1 , 2 , 7 ) . {\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.} In

1802-425: A corresponding bound vector, in this sense, whose initial point has the coordinates of the origin O = (0, 0, 0) . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x -axis. This coordinate representation of free vectors allows their algebraic features to be expressed in

1908-749: A major port in 1886 when the port opened to trade, following the initial construction of the dock basin in 1882. In 1810 or 1811 seven Martello Towers were built along the shore, of which 4 (Tower P on Wireless Green off Old Fort Road is the home of the local CoastWatch group, Q Tower in the town, and two more towards the Deben mouth) survive. Q Tower was the HQ of the Harwich-Ipswich-Martlesham Heath anti-aircraft guns between 1941 and 1945 (earlier it had been in Landguard Fort). At

2014-417: A period of more than 200 years. About a dozen people contributed significantly to its development. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence . Working in a Euclidean plane, he made equipollent any pair of parallel line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in

2120-425: A space with no notion of length or angle. In physics, as well as mathematics, a vector is often identified with a tuple of components, or list of numbers, that act as scalar coefficients for a set of basis vectors . When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but

2226-484: A special kind of vector space called Euclidean space . This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric , spatial , or Euclidean vectors. A Euclidean vector may possess a definite initial point and terminal point ; such

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2332-406: A vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set. The decomposition or resolution of a vector into components is not unique, because it depends on

2438-726: A vector is one type of tensor . In pure mathematics , a vector is any element of a vector space over some field and is often represented as a coordinate vector . The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction". Vectors are usually denoted in lowercase boldface, as in u {\displaystyle \mathbf {u} } , v {\displaystyle \mathbf {v} } and w {\displaystyle \mathbf {w} } , or in lowercase italic boldface, as in

2544-494: A vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the flights of an arrow from the back. In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n -dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system . The endpoint of a vector can be identified with an ordered list of n real numbers ( n - tuple ). These numbers are

2650-853: A vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame ). The following section uses the Cartesian coordinate system with basis vectors e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)} and assumes that all vectors have

2756-471: Is Minkowski space (which is important to our understanding of special relativity ). However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of vector spaces (for free vectors) and affine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics , where many quantities of interest can be considered vectors in

2862-416: Is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ s of 4 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to

2968-417: Is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as A B ⟶ {\displaystyle {\stackrel {\longrightarrow }{AB}}} or AB . In German literature, it was especially common to represent vectors with small fraktur letters such as

3074-596: Is a scheduled ancient monument and visitor attraction with a nearby nature reserve . At the opposite end of the town is Felixstowe Ferry Golf Club , which is amongst the oldest in the UK, having been established in 1881. The Rt. Hon. Arthur Balfour , Captain of the Golf Club in 1889, became captain of the R&;A in 1894 and British Prime Minister from 1902 to 1905. Felixstowe has a recently refurbished sandy beach south from

3180-466: Is a walkway and is the start of the 50-mile Suffolk Coast Path . At low tide from Jacob's Ladder, it is possible to glimpse the seaweed-covered remains of a Roman fort , which could possibly be the place of Dommoc , in the water about 50 yards from the coast, where it suggested St Felix first landed in England. Perhaps the most striking building on the front is Harvest House. Originally built as

3286-678: Is based in Felixstowe Town Hall , on the seafront at Undercliff Road West. Felixstowe lies within the Suffolk Coastal parliamentary constituency; Jenny Riddell-Carpenter of the Labour Party was elected as the Member of Parliament in the 2024 General Election . In 1894 an urban district called "Felixstowe and Walton" was formed which contained the parishes of "Felixstowe" and "Walton", in 1889 it became part of

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3392-465: Is essentially the modern system of vector analysis. In 1901, Edwin Bidwell Wilson published Vector Analysis , adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus. In physics and engineering , a vector is typically regarded as a geometric entity characterized by a magnitude and a relative direction . It is formally defined as

3498-476: Is from the Anglo-Saxon or Old English name "Filicia" and "stōw", meaning a place of location. Literally Filicia's place. The earliest recorded names "Filchestou" from 1254, and "Filchestowe" in 1291 support this idea. It is possible that the later reworking of Filicia/Filche was made with the intention of referencing Felix of Burgundy. The old Felixstowe hamlet was centred on a pub and church, having stood on

3604-636: Is generally considered part of Felixstowe due to modern expansion. Felixstowe is situated at the tip of the Colneis peninsula, and was in the ancient Colneis Hundred of Suffolk. The Felixstowe area as a whole provided a linchpin in England's defence, as proved in 1667 when Dutch soldiers landed near the Fludyers area and tried (unsuccessfully) in the Battle of Landguard Fort to capture Landguard Fort due to strategic location. The town only became related to

3710-431: Is often presented as the standard Euclidean space of dimension n . This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} More precisely, given such a Euclidean space, one may choose any point O as an origin . By Gram–Schmidt process , one may also find an orthonormal basis of

3816-660: Is owned by Hutchison Port Holdings Ltd with additional land on the peninsula owned by Trinity College, Cambridge . The port has its own Police Authority , which also currently has jurisdiction over the area local to the port, with permission from Suffolk Constabulary 's Chief Constable. Alongside the Port Police, they also have their own joint ambulance and fire service; one of the port ambulances, call sign Alpha 1 , can also come off port to attend 999 emergencies in Felixstowe. The Harwich Harbour Ferry operates between

3922-541: Is shown, and not Felixstowe, which at the time held little more than a few houses scattered over the cliff tops. Walton was a settlement on the River Orwell and in 1844 had a population of 907 compared to the relatively small Felixstowe Parish holding only 502 people. Walton had always preceded Felixstowe as a settlement as seen by the presence of Walton Castle , built by the Romans in the 3rd century, but today Walton

4028-563: Is the largest container port in the United Kingdom. Felixstowe is approximately 72 miles (116 km) northeast of London. There are competing theories as to how the name of Felixstowe arose. One is that the town is named after Felix of Burgundy , a saint and the first bishop of the East Angles in the seventh century, although this is unlikely as the name Felixstowe is not recorded for almost 900 years. An alternative etymology

4134-568: Is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ( A , B ) ) and same direction (e.g., the direction from A to B ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars , which have no direction. For example, velocity , forces and acceleration are represented by vectors. In modern geometry, Euclidean spaces are often defined from linear algebra . More precisely,

4240-413: The n -tuple of its Cartesian coordinates, and every vector to its coordinate vector . Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons . If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F

4346-599: The abdication crisis in the same year. Most of the south-western area of Felixstowe Urban District, between the Dock, Landguard Point, and Manor Road, was occupied by the Navy, RAF and Army. with Landguard Fort and several ruined gun emplacements and bunkers a reminder of the 1939–1945 era. It was the first base from which Second World War German E-boats and coastal convoys were systematically attacked—by flotilla led by Lt-Commanders Howes, Dickens, Hichens and Trelawney. Felixstowe

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4452-487: The coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system , and are typically called the scalar components (or scalar projections ) of the vector on the axes of the coordinate system. As an example in two dimensions (see figure), the vector from the origin O = (0, 0) to the point A = (2, 3) is simply written as a = ( 2 , 3 ) . {\displaystyle \mathbf {a} =(2,3).} The notion that

4558-417: The dot product . This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, R n {\displaystyle \mathbb {R} ^{n}} is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}}

4664-417: The electric and magnetic field , are represented as a system of vectors at each point of a physical space; that is, a vector field . Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors. In the Cartesian coordinate system , a bound vector can be represented by identifying

4770-396: The hat symbol ^ {\displaystyle \mathbf {\hat {}} } typically denotes unit vectors ). In this case, the scalar and vector components are denoted respectively a x , a y , a z , and a x , a y , a z (note the difference in boldface). Thus, a = a x + a y +

4876-448: The real line , Hamilton considered the vector v to be the imaginary part of a quaternion: The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion. Several other mathematicians developed vector-like systems in

4982-491: The 19th century) as equivalence classes under equipollence , of ordered pairs of points; two pairs ( A , B ) and ( C , D ) being equipollent if the points A , B , D , C , in this order, form a parallelogram . Such an equivalence class is called a vector , more precisely, a Euclidean vector. The equivalence class of ( A , B ) is often denoted A B → . {\displaystyle {\overrightarrow {AB}}.} A Euclidean vector

5088-498: The Felix Hotel, it then became Fisons ' headquarters. It has now been converted to apartments for residents of retirement age, many with fabulous views over Felixstowe. This was built on the sea front, had 19 beds and was open from about 1902 until the 1970s. It was run by The London Hospital to provide a space where patients could recuperate by the sea. Nurses were also sometimes sent there from The London. After its closure

5194-638: The Felixstowe Museum. There are a number of hotels and guesthouses, self-catering properties and camping/caravanning sites catering for visitors. Felixstowe Rugby Union Football Club was founded in 1930 and plays in the Eastern Counties Leagues. Felixstowe Hockey Club has three men's teams and three ladies' team. The men's first team play in Eastern Division 2 North East. The club was voted England Hockey's "club of

5300-671: The MTB base and later the container port. On 11 August 1919, the Felixstowe Fury sideslipped and crashed into the sea 500 yards offshore soon after take-off while on a test flight. It was preparing for an 8,000-mile flight to Cape Town, South Africa. The wireless operator, Lt. MacLeod, was killed, and the six passengers were rescued. The wreckage was towed ashore. Wallis Simpson stayed in Felixstowe in 1936 in order to claim residence for her divorce from Ernest Simpson so that she could marry Edward VIII . The divorce and marriage sparked

5406-582: The North Hinder Light Vessel, a Dutch maintained light ship 55 miles from Felixstowe and the Hook of Holland as a centre point. An octagonal figure was drawn with eight arms radiating out from a distance of 30 miles from the centre. A set of circumvential lines then joined the radial arms at 10, 20 and 30 miles making eight sectors, each sector divided into three sections. As the patrolling flying boat flew up and down each sector line ,

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5512-648: The Royal Navy. In 1953, at least 48 people died in the town during the North Sea flood . In 1993 'Fast Eddie Maher', a security van driver, stole £1.2m cash from outside Lloyds bank. He went on the run and was eventually arrested in 2012. Landguard Fort, originally known as Langer Fort, is on the site of the last opposed invasion of England in 1667, and the first land battle of the Duke of York and of Albany 's (later James II & VII ) Marines. The current fort

5618-674: The Second World War. It was also used as a base by the Schneider Trophy team, the High Speed Flight . RAF Felixstowe closed 21 June 1962. Felixstowe Felixstowe ( / ˈ f iː l ɪ k s t oʊ / FEE -lic-stoh ) is a port town and civil parish in the East Suffolk district, in the county of Suffolk , England. The estimated population in 2017 was 24,521. The Port of Felixstowe

5724-543: The Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B . Many algebraic operations on real numbers such as addition , subtraction , multiplication , and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity , associativity , and distributivity . These operations and associated laws qualify Euclidean vectors as an example of

5830-424: The United Kingdom. As the name implies, the unit designed seaplanes and flying boats . These were generally known by the Felixstowe name although, apart from the prototypes, these flying boats were built by aircraft manufacturers such as Short Brothers , Dick, Kerr & Co. and Phoenix Dynamo Manufacturing Company (the latter two forming part of English Electric in 1918–1919). Upon Porte's recommendation,

5936-817: The View Point (near Landguard Fort) in Felixstowe and Ha'penny Pier, Harwich and Shotley Point throughout the summer. The Bawdsey Ferry crosses the River Deben from Felixstowe Ferry . Felixstowe railway station is the eastern terminus of the passenger service to Ipswich , via the Felixstowe Branch Line ; the line itself branches before the station, the other line goes on to the Port of Felixstowe . Local bus services are operated by First Eastern Counties , with routes to Ipswich and Woodbridge . The National Express coach service 481 provides

6042-530: The Zeppelins; the aircraft, two Bristol Scouts , took off from a short improvised runway on the forward deck. The station also serviced aircraft of the carriers Engadine and Campania . On 24 April 1916 trials were run in conjunction with the Submarine Service at Parkeston Quay to test the carriage and launching of 2 Sopwith Schneider seaplanes carried on the deck of submarine E22 . E22

6148-447: The administrative county of East Suffolk . In 1914 Walton parish was merged with Felixstowe and the urban district was renamed to just "Felixstowe". On 1 April 1974 the district was abolished and became part of Suffolk Coastal in the non-metropolitan county of Suffolk. A successor parish was formed covering the same area as the former district and its parish. In 2019 Felixstowe became part of East Suffolk district. Landguard Fort

6254-820: The aircraft, experiments were carried out in the launch and retrieval of flying boats and bi-planes from specially designed lighters towed behind destroyers of the Harwich Force . On the formation of the Royal Air Force 1 April 1918, the unit was renamed the Seaplane Experimental Station, Felixstowe and disbanded in June 1919. The base and its facilities were later used by the Marine Aircraft Experimental Establishment from 1 April 1924 until

6360-434: The area of the modern day docks at a small pier popular with pleasure boats, and with a paddle steamer link to London. A dock next to the pier was approved in 1879. Felixstowe is Britain's largest container port. The main navigation channel is dredged to 47 + 1 ⁄ 2 feet (14.5 m) below chart datum and a depth of up to 49 feet (15 m) alongside the quay. Felixstowe boasts deep-water able to accommodate

6466-405: The area was surveyed twice on any patrol and two sectors of the web could be patrolled in under five hours . A flying boat would take off from Felixstowe and head for the North Hinder Light Vessel then fly along a sector line, determined by previous instructions gained from wireless plots, and then along the patrol lines of the sector. "Web" patrols commenced 13 April 1917. To increase the range of

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6572-534: The associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector O P → . {\displaystyle {\overrightarrow {OP}}.} These choices define an isomorphism of the given Euclidean space onto R n , {\displaystyle \mathbb {R} ^{n},} by mapping any point to

6678-433: The basis has, so the components of the vector must change to compensate. The vector is called covariant or contravariant , depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on

6784-442: The choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including

6890-441: The cliffs above this area sits Harvest House, which was built as The Felix Hotel in 1903, and was known as "the millionaire's hotel", because of the gentry and royal visitors it attracted. The Floral Hall was built in 1909, and subsequently became known as The Spa Pavilion. Felixstowe remained a fashionable seaside destination until the late 1930s. In April 1914 as part of the suffragette bombing and arson campaign Bath Hotel, which

6996-441: The complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs , who was exposed to quaternions through James Clerk Maxwell 's Treatise on Electricity and Magnetism , separated off their vector part for independent treatment. The first half of Gibbs's Elements of Vector Analysis , published in 1881, presents what

7102-410: The coordinates of its initial and terminal point. For instance, the points A = (1, 0, 0) and B = (0, 1, 0) in space determine the bound vector A B → {\displaystyle {\overrightarrow {AB}}} pointing from the point x = 1 on the x -axis to the point y = 1 on the y -axis. In Cartesian coordinates, a free vector may be thought of in terms of

7208-433: The direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, a vector perpendicular to the plane of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates

7314-411: The geometrical and physical settings, it is sometimes possible to associate, in a natural way, a length or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an angle between two vectors. If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives

7420-556: The harbour entrance, and a radio and radar station. Landguard Fort is in the care of English Heritage , and is managed by the Landguard Fort Trust to make it accessible to the public. A museum telling the story of Felixstowe, with a reference library, historic maps, photo archive and fourteen rooms of artefacts from Roman finds, the Martello towers , military social and domestic history through two world wars and into

7526-595: The home became the Herman de Sterne Centre. Following a severe fire it was demolished in 2006. Two matrons who had trained at The London Hospital, under Eva Luckes were: Felixstowe draws tourist visitors and has traditional seaside offerings such as Felixstowe Pier and The Seafront Gardens. There is also the Forum Amusement Centre, Ocean Boulevard and Manning's Amusements. Visitors can also see Landguard Fort, Landguard Nature Reserve, Felixstowe Ferry and

7632-435: The intuitive interpretation as vectors of unit length pointing up the x -, y -, and z -axis of a Cartesian coordinate system , respectively. In terms of these, any vector a in R can be expressed in the form: a = ( a 1 , a 2 , a 3 ) = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) +

7738-406: The length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors . The vector concept, as it is known today, is the result of a gradual development over

7844-424: The local council, now managed by a contractor. Felixstowe has a Royal Air Force Air Cadet Squadron, 356 (Felixstowe) Squadron ATC. Cobbolds Point takes its name from Felix Cobbold , who built Felixstowe Lodge, now known as Cranmer House, and associated sea wall at the bottom of Maybush Lane in 1885. It is a Grade II- listed mock Elizabethan mansion by Thomas Cotman . Its listing describes it as "a fine house of

7950-520: The middle of the nineteenth century, including Augustin Cauchy , Hermann Grassmann , August Möbius , Comte de Saint-Venant , and Matthew O'Brien . Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work

8056-483: The more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important role in physics : the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement ), their magnitude and direction can still be represented by

8162-614: The new millennium is managed by volunteers from the Felixstowe History and Museum Society. It is located in the old submarine mining establishment building at the Landguard Peninsula, between the Fort and Port. The pier was opened in 1906, rebuilt in late 2017, and re-opened in 2018. During the Second World War the majority of the pier, at the time one of the longest in the country and complete with its own train,

8268-405: The origin as a common base point. A vector a will be written as a = a 1 e 1 + a 2 e 2 + a 3 e 3 . {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.} Two vectors are said to be equal if they have

8374-703: The original bungalow that forms the lower two floors of Cotman Lodge care home. He was the nephew of John Sell Cotman , the famous Norwich water-colour artist. The Art Deco style of architecture has been used for some buildings. The Cavendish Hotel in Sea Road, demolished in 1988, was in this style. There is another example in Undercliff Road West, which has been home to the Felixstowe Tourist Information Centre. From Brackenberry Fort to Felixstowe Ferry there

8480-488: The other hand, have units of one-over-distance such as gradient . If you change units (a special case of a change of basis ) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1  K /m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors ). Tensors are another type of quantity that behave in this way;

8586-408: The parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the exterior product , which (among other things) supplies an algebraic characterization of the area and orientation in space of the n -dimensional parallelotope defined by n vectors. In a pseudo-Euclidean space , a vector's squared length can be positive, negative, or zero. An important example

8692-449: The perimeter of the main building. The sole remaining railway station Felixstowe , was opened in 1898. The well-preserved station building now houses a supermarket and shops. In its prime the railway station saw more than twenty services a day and is now served by an hourly service to Ipswich . The station now has only one platform, which has been created from the far end of one of the original platforms. Felixstowe Beach railway station

8798-552: The period reflecting the wealth of this important Suffolk family of brewers". However Felix was involved in the banking and insurance interests of the family rather than its brewing assets. Cotman, the architect, designed many of the most famous buildings in Felixstowe including the Railway Station, Harvest House (Felix Hotel), the Orwell and Bath Hotels, Barclays and Lloyds Banks plus many others. He also designed and lived in

8904-438: The pier, and a stoney beach north of the pier. A Victorian promenade runs along part of the beach, from the nature reserve in the southwest to Cobbolds Point (Maybush Lane in east), with traditional beach huts along most of that length. An amusement arcade with snooker halls and food outlets occupies the southern end. The pier, incorporating a cafe and amusement arcade, stands before a leisure centre, with swimming pool, owned by

9010-434: The plane, and thus erected the first space of vectors in the plane. The term vector was introduced by William Rowan Hamilton as part of a quaternion , which is a sum q = s + v of a real number s (also called scalar ) and a 3-dimensional vector . Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex numbers use an imaginary unit to complement

9116-596: The rules of vector addition. An example is velocity , the magnitude of which is speed . For instance, the velocity 5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positive y -axis as 'up'). Another quantity represented by a vector is force , since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement , linear acceleration, angular acceleration , linear momentum , and angular momentum . Other physical vectors, such as

9222-563: The same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram . If the Euclidean space is equipped with a choice of origin , then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications. In classical Euclidean geometry (i.e., synthetic geometry ), vectors were introduced (during

9328-699: The same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors a = a 1 e 1 + a 2 e 2 + a 3 e 3 {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} and b = b 1 e 1 + b 2 e 2 + b 3 e 3 {\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} are equal if

9434-575: The season" in 2014 The town's only senior football team, Felixstowe & Walton United , compete in the Isthmian North Division (Step 4). Felixstowe and Walton also have junior teams as well as a reserve team. Euclidean vector A vector is what is needed to "carry" the point A to the point B ; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around

9540-525: The site since long before the Norman conquest of England . The early history of Felixstowe, including its Roman , Anglo-Saxon , Norman and medieval defences, is told under the name of Walton , because the name Felixstowe was given retrospectively, during the 13th century, to a place which had expanded to a form beyond the boundaries of Walton alone. In the Domesday Book , for instance, only Walton

9646-567: The station was initially equipped with Curtiss flying boats. He improved their hull designs, before developing the Felixstowe flying boats from those experiments. Many Felixstowe boats were built under licence in the USA. The craft were flown on long-range patrols to spot the German High Seas Fleet and Zeppelins , with many based at RNAS Felixstowe. A seaplane carrier, HMS Vindex based at Felixstowe, planned to operate against

9752-412: The tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation O A → {\displaystyle {\overrightarrow {OA}}} is usually deemed not necessary (and is indeed rarely used). In three dimensional Euclidean space (or R ), vectors are identified with triples of scalar components: a = (

9858-482: The turn of the century, tourism increased, and a pier was constructed in 1905, some of which is partially functional to this day as an amusement arcade . Indeed, during the late Victorian period (after circa 1880) it became a fashionable resort, a trend initiated by the opening of Felixstowe Town railway station , the pier and a visit by the German imperial family in 1891. It was in this buoyant period that Felixstowe

9964-487: The two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} equipped with

10070-702: The unit vectors of a cylindrical coordinate system ( ρ ^ , ϕ ^ , z ^ {\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} } ) or spherical coordinate system ( r ^ , θ ^ , ϕ ^ {\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}} ). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. The choice of

10176-556: The world's latest generation of deep-draughted ultra post-Panamax vessels. There is a continuous quay of 2.4 km, equipped with 25 ship-to-shore gantry cranes. The town has road links to the Midlands , via the A14 , and to London , via the A12 road . The single-track railway line to Ipswich has been upgraded recently to allow larger containers; many more are now transported by rail. The port

10282-887: Was also HQ of the Harwich Harbour coast and anti-aircraft defences, and accommodated the RAF's 26th Marine Craft (Air-Sea Rescue) Unit. In 1944 the piers near the Dock were used to load troops, tanks and vehicles onto the British and American landing craft of "Force L", which reinforced the Normandy Invasion on its first and second days. In 1945 the German naval commanders in Occupied Holland arrived in E-boats at Felixstowe Dock to surrender their boats and charts to

10388-524: Was announced in April 1913, to be developed along the lines of the station already established on the Isle of Grain . The unit at Felixstowe was commissioned 5 August 1913 on the River Orwell at Landguard under the command of Captain C. E. Risk, RM as Seaplanes, Felixstowe followed by Lieutenant C. E. H. Rathborne, RN in 1914 and Lieutenant-Commander John Cyril Porte , RN 1915. RNAS Felixstowe

10494-428: Was built in the 18th century, and modified in the 19th century with substantial additional 19th/20th century outside batteries. The Fort hosts regular military re-enactments, including Darell's Day , which is a celebration of the last invasion, children's events and open-air theatre. In the two world wars the Fort was variously the HQ of the Harwich Harbour coast and anti-aircraft defences, the signal/control station for

10600-609: Was created soon after the outbreak of World War I following the formation of the Royal Naval Air Service 1 July 1914, from the Naval Wing of the Royal Flying Corps (RFC). Three large hangars 300 feet long and 200 feet wide, with slipways were built by Norwich engineers Boulton & Paul and camouflage paint schemes applied; the base would become the largest operational seaplane station in

10706-405: Was demolished in 2004, despite a storm of protest from many local people keen on saving the historic building which the council had branded as "unsafe". The station was originally opened in 1877 and was used continuously until 1959, after which it was the site of a small printers for many years until its demolition. From 1877 until 1951 there was also Felixstowe Pier railway station , sited inside

10812-407: Was largely neglected until the 1870s. Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. In 1878, Elements of Dynamic was published by William Kingdon Clifford . Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from

10918-475: Was popular with elite vistors, was burnt down. Felixstowe played an important role in both world wars; in the first as Royal Naval Air Service and RAF seaplane base, and in the second as the Coastal Forces MTB, MGB and ML base HMS Beehive. Between the wars the seaplane station housed the RAF experimental establishment which tested seaplanes and flying boats. Its sheds and piers were incorporated in

11024-484: Was purposely demolished by the Royal Engineers to prevent it from being used as an easy landing point for enemy troops. After the war the damage was not repaired and the pier never regained its original length. The pier in its current incarnation features an amusement hall with a gambling section, traditional fish and chips and ice cream kiosks, and a restaurant/bar with indoor and outdoor seating. The deck spans

11130-434: Was sunk the following day off Great Yarmouth by German U-boat SM UB-18 . To begin with the flying boats had little success against U-boats until the introduction of the " Spider web " system of patrolling. The patrols capitalised on the practice of U-boats signalling by wireless their homing position, which could be picked up by wireless stations at Hunstanton , Lowestoft and Birchington . The "Spider Web" used

11236-406: Was the first British town to adopt beach huts as stationary permanent structures (rather than wheeled bathing machines) and there are photographic records of beach huts at the "spa area" in Felixstowe dating back to 1895. It is therefore known as the home to British beach huts. Some of the beach huts in the spa area (a conservation area) date back to c.1900, so are probably the original beach huts. On

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