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The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.
The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as = ^ where is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as [2] = ‖ ‖ = ^ where the operator ⋅ denotes a dot product, ‖a‖ is the length of ...
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
When θ is a right angle, the versor is a right versor: its scalar part is zero and its vector part v is a unit vector in . Thus the right versors extend the notion of imaginary units found in the complex plane , where the right versors now range over the 2-sphere S 2 ⊂ R 3 ⊂ H {\displaystyle \mathbb {S} ^{2}\subset \mathbb {R} ^{3}\subset ...
This matrix equation relates the scalar components of a in the n basis (u,v, and w) with those in the e basis (p, q, and r). Each matrix element c jk is the direction cosine relating n j to e k. [19] The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. [19] Therefore,
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Texas Lt. Gov. Dan Patrick said U.S.-owned border wall materials, which were available for sale, were pulled from an Arizona auction at the government's request. The Lonestar State had shown ...
Given a 3 × 3 rotation matrix R, a vector u parallel to the rotation axis must satisfy =, since the rotation of u around the rotation axis must result in u. The equation above may be solved for u which is unique up to a scalar factor unless R = I. Further, the equation may be rewritten