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A prototype of a vector is a directed line segment AB see Figure 1 that can be thought to represent the displacement of a particle from its initial position A to a new position B.
To distinguish vectors from scalars it is customary to denote vectors by boldface letters.
Thus the vector AB in Figure 1 can be denoted by a and its length or magnitude by a. In many problems the location of the initial point of a vector is immaterial, so that two vectors are regarded as equal if they have the same length and the same direction.
This construction of the sum, c, of a and b yields the same result as the parallelogram law in which the resultant c is given by the diagonal AC of the parallelogram constructed on vectors AB and AD as sides. Also, it is easy to show that the associative law is valid, and hence the parentheses in 2 can be omitted without any ambiguities.
If s is a scalar, sa or as is defined to be a vector whose length is s a and whose direction is that of a when s is positive and opposite to that of a if s is negative.
Thus, a and -a are vectors equal in magnitude but opposite in direction. The foregoing definitions and the well-known properties of scalar numbers represented by s and t show that Inasmuch as the laws 12and 3 are identical with those encountered in ordinary algebra, it is quite proper to use familiar algebraic rules to solve systems of linear equations containing vectors.
This fact makes it possible to deduce by purely algebraic means many theorems of synthetic Euclidean geometry that require complicated geometric constructions. The multiplication of vectors leads to two types of products, the dot product and the cross product.
The associative, commutative, and distributive laws of elementary algebra are valid for the dot multiplication of vectors. Also, since rotation from b to a is opposite to that from a to b, Figure 2: Since empirical laws of physics do not depend on special or accidental choices of reference frames selected to represent physical relations and geometric configurations, vector analysis forms an ideal tool for the study of the physical universe.
The introduction of a special reference frame or coordinate system establishes a correspondence between vectors and sets of numbers representing the components of vectors in that frame, and it induces definite rules of operation on these sets of numbers that follow from the rules for operations on the line segments.
If some particular set of three noncollinear vectors termed base vectors is selected, then any vector A can be expressed uniquely as the diagonal of the parallelepiped whose edges are the components of A in the directions of the base vectors. In common use is a set of three mutually orthogonal unit vectors i.
In this system the expression takes the form Figure 3: Also, the dot product can be written since The use of law 6 yields for so that the cross product is the vector determined by the triple of numbers appearing as the coefficients of i, j, and k in 9. Such rephrasing suggests a generalization of the concept of a vector to spaces of dimensionality higher than three.
For example, the state of a gas generally depends on the pressure p, volume v, temperature T, and time t. A quadruple of numbers p,v,T,t cannot be represented by a point in a three-dimensional reference frame. But since geometric visualization plays no role in algebraic calculations, the figurative language of geometry can still be used by introducing a four-dimensional reference frame determined by the set of base vectors a1,a2,a3,a4 with components determined by the rows of the matrix A vector x is then represented in the form so that in a four-dimensional spaceevery vector is determined by the quadruple of the components x1,x2,x3,x4.
A particle moving in three-dimensional space can be located at each instant of time t by a position vector r drawn from some fixed reference point O.
Since the position of the terminal point of r depends on time, r is a vector function of t. Its components in the directions of Cartesian axes, introduced at O, are the coefficients of i, j, and k in the representation If these components are differentiable functions, the derivative of r with respect to t is defined by the formula which represents the velocity v of the particle.
The Cartesian components of v appear as coefficients of i, j, and k in The rules for differentiating products of scalar functions remain valid for derivatives of the dot and cross products of vector functions, and suitable definitions of integrals of vector functions allow the construction of the calculus of vectors, which has become a basic analytic tool in physical sciences and technology.
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The federal Fair Labor Standards Act (FLSA) and the laws in most states require that employers keep accurate records of hours worked and wages paid to nonexempt employees. Records of hours worked can be tracked using a variety of methods, including handwritten time cards, time clocks, or Web clocks.
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Records of hours worked can be tracked using a variety of methods, including handwritten time cards, time clocks, or Web clocks, or.