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The scalar product of a force F and the velocity v of its point of application defines the power input to a system at an instant of time. Integration of this power over the trajectory of the point of application, C = x ( t ) , defines the work input to the system by the force.
In mathematics, the dot product or scalar product [note 1] 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.
The result, div F, is a scalar function of x. Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system . However the above definition is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to ...
In the image, the vector F 1 is the force experienced by q 1, and the vector F 2 is the force experienced by q 2. When q 1 q 2 > 0 the forces are repulsive (as in the image) and when q 1 q 2 < 0 the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.
extensive, bivector or scalar Centrifugal force: F c: Inertial force that appears to act on all objects when viewed in a rotating frame of reference: N⋅rad = kg⋅m⋅rad⋅s −2: L M T −2: bivector Crackle: c →: Change of jounce per unit time: the fifth time derivative of position m/s 5: L T −5: vector Current density: J →
Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity. Mechanical power is also described as the time derivative of work.