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In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O , and its direction represents the angular orientation with respect to given reference axes.
An absolute bearing is measured with a bearing compass. The measurement of absolute bearings of fixed landmarks and other navigation aids is useful for the navigator because this information can be used on the nautical chart together with simple geometrical techniques to aid in determining the position of the vessel.
where | g | is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor . These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components is given by:
Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force acting on that body M about that point: [1] [4] [5]
where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame. [note 3] Taking the differential of the first of the two equations above, we have, ′ =, and what may seem like the obvious [note 4] statement that ′ =, we have:
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another ... — Isaac Newton These notions imply that absolute space and time do not depend upon physical events, but are a backdrop or stage setting within which physical phenomena occur.
Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.. In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. [1]