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  2. Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Euclidean_space

    The inner product of a Euclidean space is often called dot product and denoted x ⋅ y. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly ...

  3. Euclidean vector - Wikipedia

    en.wikipedia.org/wiki/Euclidean_vector

    The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as

  4. Analytic geometry - Wikipedia

    en.wikipedia.org/wiki/Analytic_geometry

    In three dimensions, distance is given by the generalization of the Pythagorean theorem: = + + (), while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by [22] = ‖ ‖ ‖ ‖ ⁡, where θ is the angle between A and B.

  5. Vector space - Wikipedia

    en.wikipedia.org/wiki/Vector_space

    The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G.

  6. Dot product - Wikipedia

    en.wikipedia.org/wiki/Dot_product

    In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).

  7. Real coordinate space - Wikipedia

    en.wikipedia.org/wiki/Real_coordinate_space

    Any Euclidean n-space has a coordinate system where the dot product and Euclidean distance have the form shown above, called Cartesian. But there are many Cartesian coordinate systems on a Euclidean space. Conversely, the above formula for the Euclidean metric defines the standard Euclidean structure on R n, but it is

  8. Plane (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Plane_(mathematics)

    A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.

  9. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a ...