Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Unit vectorIn mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in (pronounced "v-hat"). The term direction vector, commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere.
Vector spaceIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
Four-vectorIn special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (1/2,1/2) representation. It differs from a Euclidean vector in how its magnitude is determined.
AccelerationIn mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes: the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force; that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass.
Dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. 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 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).
Triple productIn geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
Inner product spaceIn mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates.
RotationRotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis.
KinematicsKinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system.
Inverse kinematicsIn computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics.
Robot kinematicsIn robotics, robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation. Robot kinematics studies the relationship between the dimensions and connectivity of kinematic chains and the position, velocity and acceleration of each of the links in the robotic system, in order to plan and control movement and to compute actuator forces and torques.
Forward kinematicsIn robot kinematics, forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters. The kinematics equations of the robot are used in robotics, computer games, and animation. The reverse process, that computes the joint parameters that achieve a specified position of the end-effector, is known as inverse kinematics.
Vector calculusVector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations.
Rotation (mathematics)Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.
Tangent vectorIn mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the set of germs at .
Kinematic chainIn mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system. As the word chain suggests, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is the kinematic model for a typical robot manipulator. Mathematical models of the connections, or joints, between two links are termed kinematic pairs.
Scalar multiplicationIn mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors.
Rotations in 4-dimensional Euclidean spaceIn mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise. A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation.
Cartesian coordinate systemIn geometry, a Cartesian coordinate system (UKkɑːrˈtiːzjən, USkɑːrˈtiʒən) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where they meet is called the origin and has (0, 0) as coordinates.
