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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 ...
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 ...
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,
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.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
where v x, v y, and v z are the scalar components of v. Scalar components may be positive or negative; the absolute value of a scalar component is its magnitude. Scalar components may be positive or negative; the absolute value of a scalar component is its magnitude.
For any vector space V, the projection X × V → X makes the product X × V into a "trivial" vector bundle. Vector bundles over X are required to be locally a product of X and some (fixed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of π to π −1 (U) is isomorphic [nb 11] to the trivial bundle ...