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Vector projection of a on b (a 1), and vector rejection of a from b (a 2). In mathematics, the scalar projection of a vector on (or onto) a vector , also known as the scalar resolute of in the direction of , is given by:
The scalar projection is defined as [2] = ‖ ‖ = ^ where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b ...
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 modified Rodrigues vector is a stereographic projection mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane. The projection of the opposite quaternion −q results in a different modified Rodrigues vector p s than the projection of the original quaternion q.
where a 1, a 2, a 3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z (see figure), while a 1, a 2, a 3 are the respective scalar components (or scalar projections).
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group = /, where GL(V) is the general linear group of invertible linear transformations of V over F, and F ∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see ...
The identity map and the scalar projection operator are outermorphisms. Versors. A rotation of a vector by a rotor is given by = with outermorphism _ =. We check that this is the correct form of the outermorphism.
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]