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In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
Three line segments with the same direction. In geometry, direction, also known as spatial direction or vector direction, is the common characteristic of all rays which coincide when translated to share a common endpoint; equivalently, it is the common characteristic of vectors (such as the relative position between a pair of points) which can be made equal by scaling (by some positive scalar ...
This matrix equation relates the scalar components of a in the n basis (u,v, and w) with those in the e basis (p, q, and r). Each matrix element c jk is the direction cosine relating n j to e k. [19] The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. [19] Therefore,
The Euclidean distance is the prototypical example of the distance in a metric space, [11] and obeys all the defining properties of a metric space: [12] It is symmetric, meaning that for all points and , (,) = (,). That is (unlike road distance with one-way streets) the distance between two points does not depend on which of the two points is ...
Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them. [20]
A metric or distance function is a function d which takes pairs of points or objects to real numbers and satisfies the following rules: The distance between an object and itself is always zero. The distance between distinct objects is always positive. Distance is symmetric: the distance from x to y is always the same as the distance from y to x.
The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius. Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system— shown here in the mathematics convention —the sphere is adapted as a unit sphere , where the radius is set to unity and then can generally be ...
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