Vector

“ It follows that every equation of the form where is any constant and complete dyadic, may be integrated by the preceding formulæ.
note on bivector analysis. [18]
1. A vector is determined by three algebraic quantities. It often occurs that the solution of the equations by which these are to be detenmined gives imaginary values, i.e., instead of scalars we obtain biscalars, or expressions of the form where and are scalars, and It is most simple, and always allowable, to consider the vector as determined by its components parallel to a normal system of axes. ”

“ A straight line conceived in thisway as having a definite length, direction and sense, but no definite location in space, is called a vector. It may be denoted by AB (or CD) , or (when no confusion is likely to arise) simply by'AB. Thus a vector may be used to specify a displacement of translation (without rotation) of a rigid body. Again, a force acting on a particle, the velocity or momentum of a particle, the state of electric or magnetic polarization at a particular point of a medium, are examples of physical entities which are naturally represented by vectors. ”

“ By we understand the component of the vector into the direction l; by therefore the components into the axis-directions.
For a vector with components we sometimes also write .
e. If is a scalar magnitude, then we understand by the derivative with respect to time t. The letter denotes a vector with components: , or etc.
f. The expression
we call the "integral of vector over the surface ", and the magnitude
the "line integral of line s".
g. If a vector in any point of space is given, then
has everywhere a certain value, independent of the choice of coordinate system. ”

“ Scalar quantities do not involve direction. The volume of a geometrical figure, the mass and the energy of a material body, the hydrostatical pressure at a point in a fluid, and the potential at a point in space, are examples of scalar quantities.
A vector quantity has direction as well as magnitude, and is such that a reversal of its direction reverses its sign. The displacement of a point, represented by a straight line drawn from its original to its final position, may be taken as the typical vector quantity, from which indeed the name of Vector is derived. ”

“ Journal de Phys., xli, p. 51. ↑ If a and b denote two vectors, the vector whose components are () is called the vector product of a and b, and is denoted by [a, b] . Its direction is at right angles to those of a and b, and its magnitude is represented by twice the area of the triangle formed by them. ↑ If a denotes any vector, the vector whose components are is denoted by curl a. ”

“ For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r. The magnitude of the vector is the product War. ”

“ Physicists claim that there is a loss of naturalness in taking the square of a vector to be negative. In order to meet more adequately their wants, J. W. Gibbs of Yale University and A. Macfarlane of the University of Texas, have each suggested an algebra of vectors with a new notation. Each gives a definition of his own for the product of two vectors, but in such a way that the square of a vector is positive. A third system of vector analysis has been used by Oliver Heaviside in his electrical researches. ”

“ A vector of first kind is a straight line provided with a direction-sense, its components are the perpendicular projections upon the coordinate-axes. A vector of second kind is a plane surface provided with a rotation sense, its components are the perpendicular projections of the surface upon the coordinate-planes. ”

“ The second charge evidently relates to the notations and (see Nature, vol. xlvii, p. 592) . It is perfectly true that I have used a scalar and a vector connected with the linear vector operator, which, if combined, would form a quaternion. I have not. thus combined them. Perhaps Prof. Knott will say that since I use both of them it matters little whether I combine them or not. ”

“ Now what are the fundamental notions which are germane to a vector analysis? (A vector analysis is of course an algebra for vectors, or something which shall be to vectors what ordinary algebra is to ordinary quantities.) If we pass over those notions which are so simple that they go without saying, geometrical addition (denoted by ) is, perhaps, first to be mentioned. Then comes the product of the lengths of two vectors and the cosine of the angle which they include. ”

“ The ensemble of three quantities, defined for every system of Cartesian co-ordinates, and which transform as the components of an interval, is called a vector. If the three components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. ”

“ We shall see later that it is implied in Newton’s statement of his Second Law of motion. In modern language, forces are compounded by “vector-addition”; thus, if we draw in succession vectors HK>, KL> to represent P, Q, the force R is represented by the vector HL> which is the “geometric sum” of HK>, KL>.
By successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called the resultant. ”

“ Doubtless each method has advantages in certain cases, or for certain purposes. But since nothing is more simple than the definition of a linear vector function, while the definition of a quaternion is far from simple, and since in any case linear vector functions must be treated in a system of vector analysis, capacity for representing rotations does not seem to me sufficient to entitle the quaternion to a place among the fundamental and necessary notions of a vector analysis. ”

“ To show what I mean, I have only to compare Hamilton's treatment of the quaternion and of the linear vector operator with respect to notations. Since quaternions have been identified with matrices, while the linear vector operator evidently belongs to that class of multiple quantities, it seems unreasonable to refuse to the one those notations which we grant to the other. Thus, if the quaternionist has why should not the vector analyst have where represents a linear vector operator? ”

“ A radio-frequency band of 46.0 to 56.0 kilomegahertz. See also FREQUENCY, FREQUENCY BAND.
vector, n. Any quantity, such as a force, velocity, or acceleration, which has both magnitude and direction, as opposed to a SCALAR which has magnitude only. Such a quantity may be represented geometrically by an arrow of length proportional to its magnitude, pointing in the given direction.
vector, adj. A type of computerized display which consists of layers of differentiated data, each with discreet features. Individual data files can be independently manipulated. ”

“ The theory of dyadics [1] as developed in the vector analysis of 1884 must be regareded as the most important published contribution of Professor Gibbs to pure mathematics. For the vector analysis as an algebra does not fulfil the definition of the linear associative algebras of Benjamin Peirce, since the scalar product of vectores lies outside the vector domain; nor is it a geometrical analysis in the sense of Grassmann, the vector product satisfying the combinatorial law, but yielding a vector instead of a magnitude of the second order. ”

“ The properties of linear vector functions were further studied in 'Scalar invariants of two linear vector functions' (Trans. R.I.A. 1896, xxx. 709) and 'Quaternion invariants of linear vector functions' (Proc. R.I.A. 1896, iv. 1) , while the extension of the quaternion calculus to space of more than three dimensions was discussed in 'The associative algebra applicable to hyperspace' (Proc. R.I.A. 1897, v. 75) ; the algebras considered are those that are associative and distributive, and whose units satisfy equations of the same type as the units of quaternions. ”

“ In optical theories two vectors are introduced, one of which we consider as a velocity and the other as a vortex. This again is an indifferent hypothesis, since we should have arrived at the same conclusions by assuming the former to be a vortex and the latter to be a velocity. The success of the experiment cannot prove, therefore, that the first vector is really a velocity. It only proves one thing—namely, that it is a vector ”

“ If now we consider the laws of nature, we can anticipate, for two different reasons, that they must be vector or tensor equations. First, because all those physical magnitudes to which we attribute an objective existence and which enter into the expression of the laws of nature are invariants, vectors and tensors; and, secondly, because a law of nature must obviously remain unaffected by a mere change of orientation or shape of our frame of reference.
It is, indeed, evident that a mesh-system is a mere artifice of calculation. ”

“ We must show that this equation holds independently of the special choice of the Cartesian system of co-ordinates. is a vector, and therefore is also a vector, is a tensor of rank 2, is a tensor of rank 3. The second term on the left results from contraction in the indices . The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector it is necessary for to be a tensor. Then by differentiation and contraction results, and is therefore a vector, as it also is after multiplication by the reciprocal scalar . ”

“ In lecturing to students upon mathematical physics, especially in the theory of electricity and magnetism, Professor Gibbs felt, as so many other physicists in recent years have done, the desirability of a vector algebra by which the more or less complicated space relations, dealt with in many departments of physics, could be conveninently and perspicuously expressed ”

“ In the transition to the mirror image, this vector gets the opposite direction, or, as it can also be said, the opposite phase; the light vector of the other two light beams now must be changed in the same way, hence the accuracy of the above claim follows immediately.
The problem is now reduced to the two main cases, i.e. that the light vectors are everywhere perpendicular to the plane of incidence, or are everywhere located in its interior. In the course of the further investigation, we always have to think of one of these cases ”

“ But an equation which is impossible in vector analysis may be possible in bivector analysis, and in general the number of roots of an equation, or of the values of a function, will be different according as we recognise, or do not recognize, imaginary values. ”

“ It is to be observed that the term “vector ” has been used to include two distinct classes of geometrical and physical entities. The first class is typified by a displacement, or a mechanical force. A polar vector, as it is called, is a magnitude associated with a certain linear direction. This may be specified by any one of a whole assemblage of parallel lines, but the two “senses” belonging to any one of the lines are distinguished. The members of the second class, that of axial vectors, are primarily not vectors at all. An axial vector is exemplified by a couple in statics ”

“ The bivector may then be expressed in the form in which the vector factor, if we choose, may be a unit vector. In this case, we may say that the bivector has a real direction. In fact, if we express the bivector in the form the ratios of the coefficients of and which determine the direction cosines of the vector, will in this case be real. 2. The consideration that operations upon bivectors may be regarded as operations upon their biscalar x-, y- and z-components is sufficient to show the possibility of a bivector analysis and to indicate what its rules must be. ”

“ We call and the rest vectors belonging to the spacetime vector with respect to velocity , because in that coordinate system in which a point moving with velocity appears to be at rest, the first three components of and are identical with the space vectors and , which compose the transformed vector of second kind . ”

“ We remark here that ex - qmy, ey + qmx are components of the vector e + [vm] , where v is a vector in the direction of the positive Z-axis, and | v | = q, and [vm] is the vector product of v and m; similarly -qex + my, mx + qey are the components of the vector m - [ve] .
The equations 6) and 7) , as they stand in pairs, can be expressed as.
e′x′ + im′x′ = (ex + imx) cos iψ + (ey + imy) sin iψ,
e′y′ + im′y′ = - (ex + imx) sin iψ + (ey + imy) cos iψ,
e′z′ + im′z′ = e′z + imz. ”

“ In the Cartesian treatment of the subject no distinction between polar and axial vectors is necessary so long as we deal with congruent systems of co-ordinate axes. But when we pass from a right-handed to a left-handed system the formulae of transformation are different in the two cases. A polar vector (e.g. a displacement) is reversed by the process of reflection in a mirror normal to its direction, whilst the corresponding axial vector (e. a couple) is unaltered. ”

“ Physik.)
“In order to have a better knowledge about the nature of the six-vector (which is the same thing as Minkowski’s space-time vector of the 2nd kind) let us take the special case of a piece of plane, having unit area (contents) , and the form of a parallelogram, bounded by the four-vectors u, v, passing through the origin. Then the projection of this piece of plane on the xy plane is given by the projections ux, uy, vx, vy of the four vectors in the combination
φx y = uxvy - uyv{x}. ”