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A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". [4] It was first used by 18th century astronomers investigating planetary revolution around the Sun. [5] 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.
A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". [4] It was first used by 18th century astronomers investigating planetary revolution around the Sun. [5] 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.
The plane determined by the point P 0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P 0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that =
[1] [2] [3] An example of a scalar field is a weather map, with the surface temperature described by assigning a number to each point on the map. A surface wind map, [4] assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field ...
In P 2, the line between the points x 1 and x 2 may be represented as a column vector ℓ that satisfies the equations x 1 T ℓ = 0 and x 2 T ℓ = 0, or in other words a column vector ℓ that is orthogonal to x 1 and x 2. The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation ...
where is a unit vector in the direction of the line, is a point on the line, and is a scalar in the real number domain. Substituting the equation for the line into the equation for the plane gives Substituting the equation for the line into the equation for the plane gives
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.