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Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space.
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
It is an ill-posed problem as the motion happens in three dimensions (3D) but the images are a projection of the 3D scene onto a 2D plane. The motion vectors may relate to the whole image (global motion estimation) or specific parts, such as rectangular blocks, arbitrary shaped patches or even per pixel. The motion vectors may be represented by ...
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented ...
The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of 3D vector calculus that are derived as a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic field.
In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] [1] A vector space over a field F is a non-empty set V together with a binary operation and a binary function that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called ...
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):