Search results
Results from the WOW.Com Content Network
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. [30] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors.
Several methods for solving Wahba's problem are discussed by Markley and Mortari. This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the alternative formulation).
There were significant reviews given near the time of original publication. G.J.Whitrow:. Although many books have been published in recent years in which vector and tensor methods are used for solving problems in geometry and mathematical physics, there has been a lack of first-class treatises which explain the methods in full detail and are nevertheless suitable for the undergraduate student.
The decomposition or resolution [16] of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector ...
in terms of which can be separated into vector components in terms of the dual basis as a = ∑ i a i e i . {\displaystyle a=\sum _{i}a_{i}e^{i}\ .} A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. [ 29 ]
A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.
The Helmholtz decomposition in three dimensions was first described in 1849 [9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, [10] [11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. [11]