Euclidean vector

In mathematics and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, denoted by A B →. A vector is what is needed to "carry" the point A to the point B, it was first used by 18th century astronomers investigating planetary revolution around 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 and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity and distributivity; these operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined 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, their magnitude and direction can still be represented by 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 concept of vector, as we know it today, evolved over a period of more than 200 years. About a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835. Working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation, he realized an equivalence relation on the pairs of points in 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, a sum q = s + v of a Real number s 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 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 the vector of the quaternion. Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including Augustin Cauchy, Hermann Grassmann, August Möbius, Comte de Saint-Venant, Matthew O'Brien. Grassmann's 1840 work Theorie der Ebbe und Flut was the first system of spatial analysis similar to today's system and had ideas corresponding to the cross product, scalar product and vector differentiation.

Grassmann's work was 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 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, 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 is the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibb's lectures, which banished any mention of quaternions in the development of vector calculus.

In physics and engineering, a vector is regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as arrow, in a Euclidean space. In pure mathematics, a vector is defined more 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 vectors, as they are elements of a special kind of vector space called Euclidean space. This article is about vectors 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. Being an arrow, a Euclidean vector possesses a definite initial terminal point. A vector with fixed initial and terminal point is called a bound vector; when only the magnitude and direction of the vector matter the particular initial point is of no importance, the vector is called a free vector.

Thus two arrows A B → {\displaystyle {\overr

Mumbai Calling

Mumbai Calling is a British-Indian comedy series, starring Sanjeev Bhaskar, set in the fictional Teknobable call centre in Mumbai. The series was shot on location in India; the pilot first aired on ITV on 30 May 2007. The first series aired on ABC1 starting on 12 May 2009, on ITV starting on 30 May 2009. Kenny Gupta, a British Indian accountant, is sent to a call centre in Mumbai by his boss, Philip Glass, his job is to make it profitable. Kenny is joined by Glass' daughter and local call centre manager Dev Raja. After the pilot episode, Series 1 featured some major changes including replacing the character Tiffany Glass to Terri Johnson, the call centre itself looked much more modern. Teknobable Home Comforts Good Sellers Boy to Man Dating Season My Mate Mumbai All That Glitters is Not GlassThese seven new episodes were scheduled to be broadcast in a prime-time slot in ITV's 2008 Winter Schedule, however the channel changed its mind and put the episodes "on the shelf"; as a result, the new episodes were first seen on HBO India in 2008 and did not appear on UK screens until the end of May 2009.

Outsourced, a US television sitcom with a similar premise. Mumbai Calling at Mumbai Calling at British Comedy Guide Mumbai Calling on IMDb

Dobroyd Head

Dobroyd Head is a point or headland in the Northern Beaches local government area, in the suburb of Balgowlah Heights, New South Wales, Australia. It is part of the Sydney Harbour National Park, which contains examples of ecosystems at risk such as coastal heath. Tania Park is located to the immediate north-east, contains the 2MWM 90.3 transmitter. There is a lookout sited on the headland named after Arabanoo, the first Aboriginal man to live among European settlers, captured in Manly Cove in 1788. In January 1788 Captain Arthur Phillip noted Aboriginal people living in caves at what is now Wellings Reserve, Balgowlah Heights, there are a number of Aboriginal sites recorded in the area, including a midden at Reef Beach, eroded by a storm in May 1974, when human remains were exposed. What is now Dobroyd Head was named "Dobroyd Point" by Simeon Lord, a landowner in the district in the early 19th century. Dobroyd Castle, its namesake, was the home of his mother, Ann Fielden, prior to her marriage in 1764.

On his death in 1840, he gifted the land to the Crown, with a stipulation that the name must be kept. In 1871, the Secretary for Lands, John Bowie Wilson, set aside the area of 100 hectares comprising the Dobroyd headland as a defence reserve, but excluded all privately-owned lands, such as Reef Beach, Forty Baskets Beach, Grotto Point, Castle Rock and Clontarf. On 14 August 1874, prominent surveyor and hydrographer, Commander John Thomas Ewing Gowlland was drowned in an accident off the headland. In August 1963 the Manly and Pittwater Historical Society unveiled a plaque at Dobroyd Head commemorating him. In 1914, the government steamer, SS Kate, was struck and sunk by the Manly ferry Bellubera off the headland; the Dobroyd Scenic Drive, funded by the council, was opened in 1938 by Manly mayor Percy Nolan. Between 1923 and 1963, various small cabins and shacks were built around Crater Cove on the headland, they were for use as weekenders and retreats and remained occupied until the 1980s. Various subdivisions for the development of Balgowlah Heights occurred throughout the next 80 years until in 1959-1960, when Manly Council learned that land near Cutler Road and Tabalum Road was to be subdivided and objected to any development and sale of land below Cutler Road.

This movement to preserve the lands of Dobroyd Head for public recreation was led by alderman Frank Preacher, on 17 October 1960, Lands Minister Jack Renshaw met representatives of Manly Council on the site. Renshaw approved the removal of these lands from the sale of land and transferred responsibility for its preservation to Manly council. In 1975, responsibilities changed again when the area was proclaimed as part of the Sydney Harbour National Park. A 2015 article in the Manly Daily revealed that Manly Council had voted in June 1997 to erect a plaque to honour Renshaw, alderman Preacher and Manly Council's role in the preservation of the headland, but no action has since been taken to carry it out. Arabanoo lookout at Dobroyd Head - Sydney Harbour National Park