# Vector Sentence Examples

- The anomaly is then the angle BFP which the radius
**vector**makes with the major axis. - Hence equal areas are swept over by the radius
**vector**in equal times. - The problem of finding a radius
**vector**satisfying this condition is one which can be solved only by successive approximations, or tentatively. - By Kepler's second law the radius
**vector**, FP, sweeps over equal areas in equal times. - The momentum of a particle is the
**vector**obtained by multiplying the velocity by the mass in. - 2 Clerk Maxwell employed German capitals to denote
**vector**quantities. - -- o, and draw OC parallel to the
**vector-sum**. - In 1881 and 1884 he printed some notes on the elements of
**vector**analysis for the use of his students; these were never formally published, but they formed the basis of a text-book on**Vector**Analysis which was published by his pupil, E. - Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the
**vector**OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) . - On the circle, and let M be another point on the circle so related to P that the ordinate PQ moves from A to 0 in the same time as the
**vector**OM describes a quadrant. - The resultant force due to these two pointcharges must then be in the direction CP, and its value E is the
**vector**sum of the two forces along AP and BP due to the two point-charges. - (Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position
**vector**is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position**vector**of the point relative to the centre is Srg i. - This has a reciprocal Q -1= p-r = qq-1 - wp1 rq1, and a conjugate KQ (such that K[QQ'] = KQ'KQ, K[KQ] = Q) given by KQ = Kq-}-rlKp+wKr; the product QQ' of Q and Q' is app'+nqq'+w(pr'+rq'); the quasi-
**vector**RI - K) Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different sense from the rest of this article) 2(1+K)Q is Combebiac's scalar (Sp+Sq)+Combebiac's plane. - In modern language, forces are compounded by
**vector-addition**; thus, if we draw in succession~--~**vectors** **Vector**ilL which is the geometric sum of ilK, 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 -~ ~-~ -~ - In other words, a force is of the nature of a bound or localized
**vector**; it is regarded as resident in a certain line, but has no special reference to any particular point of the line. - It thus appears that an infinitesimal rotation is of the nature of a localized
**vector**, and is subject in all respects to the same mathematical Jaws as a force, conceived as acting on a rigid body. - From the equivalence of a small rotation to a localized
**vector**it follows that the rotation ~ will be equivalent to rotations E,ii, ~ about Ox, Oy, Uz, respectively, provided = le, s1 = me, i nc (I) and we note that li+,72+l~Z~i (2) - If the plane in question be chosen perpendicular to the direction of the
**vector-sum**of the given forces, the**vector-sum**of the components Q is zero, and these components are therefore equivalent to a couple (~ 4). - C Let a, ~ be the angles which AA, BB make with the direction of the
**vector-sum**, on opposite sides. - From the analogy of couples to translations which was pointed out in 7, we may infer that a couple is sufficiently represented by a free (or non-localized)
**vector**perpendicular to its plane. - The length of the
**vector**must be proportional to the moment of the couple, and its sense must be such that the sum of the moments of the two forces of the couple about it is positive. - The two forces at B will cancel, and we are left with a couple of moment P.AC in the plane AC. If we draw three
to represent these three couples, they will be perpendicular and proportional to the respective sides of the triangle ABC; hence the third**vectors****vector**is the geometric sum of the other two. - Since the given wrench can be replaced by a force acting through any assigned point P, and a couple, the locus of the null-lines through P is a plane, viz, a plane perpendicular to the
**vector**which represents the couple. - In the language of
**vector**analysis (q.v.) it is the scalar product of the**vector**representing the force and the displacement. - For, take any point 0, and construct the
**vector**~ (2) - Since the projection of a
**vector**- sum is the sum of the projections of the several, the equation (2) gives if x be the projection of 0G.**vectors** - A
**vector**OU drawn parallel to PQ, of length proportional to PQ/~I on any convenient scale, will represent the mean velocity in the interval 1t, i.e. - Indicated by this
**vector**would FIG. - Where h is constant; this shows (again) that the radius
**vector**sweeps over equal areas in equal times. - At the beginning of 13 the velocity of a moving point P was represented by a
**vector**OV drawn from a fixed origin 0. - In 1881 and 1884 he printed some notes on the elements of
**vector**analysis for the use of his students; these were never formally published, but they formed the basis of a text-book on**Vector**Analysis which was published by his pupil, E. - From the law of angular motion of the latter its radius
**vector**will run ahead of PQ near A, PQ will overtake and pass it at apocentre, and the two will again coincide at pericentre when the revolution is completed. - The angle from the pericentre to the actual radius
**vector**, and the length of the latter being found, the angular distance of the planet from the node in the plane of the orbit is found by adding to the true anomaly the distance from the node to the pericentre. - Although many pseudo-symmetric twins are transformable into the simpler form, yet, in some cases, a true polymorph results, the change being indicated, as before, by alterations in scalar (as well as
**vector**) properties. - While polysymmetry is solely conditioned by the manner in which the mimetic twin is built up from the single crystals, there being no change in the scalar properties, and the
**vector**properties being calculable from the nature of the twinning, in the case of polymorphism entirely different structures present themselves, both scalar and**vector**properties being altered; and, in the present state of our knowledge, it is impossible to foretell the characters of a polymorphous modification. - When the direction of any
**vector**quantity denoted by a symbol is to be attended to, it is usual to employ for the symbol either a block letter, as H, I, B, or a German capital, as j,, 3? - General aspects of the subject are considered under Mensuration;
**Vector**Analysis; Infinitesimal Calculus. - Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation;
**Vector**Analysis. - 656s, a way, and yp&4*t y, to write), a curve of which the radius
**vector**is proportional to the velocity of a moving particle. - Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a
**vector**representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q. - To determine the motion of a jet which issues from a vessel with plane walls, the
**vector**I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant. - Are the components of a constant
**vector**having a fixed direction; while (4) shows that the**vector**resultant of y, y, y moves as if subject to a couple of components x Wx V, x Ux W, x V-x U, (Io) and the resultant couple is therefore perpendicular to F, the resultant of x, x, x, so that the component along OF is constant, as expressed by (iii). - Well as of the body from the
**vector**OF to O'F' requires an impulse couple, tending to increase the angle F00', of magnitude, in sec. foot-pounds F.00'.sin FOO'=FVt sin (0-0), (4) equivalent to an incessant couple N=FV sin (0-0) = (F sin 0 cos 0-F cos 0 sin ¢)V = (c 2 -c i) (V /g) sin 0 cos 4) =W'(13-a)uv/g (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c 2 -c 1 is positive to set the body broadside to the advance. - Will have moved from 0 to 0', where 00' = Vt; and at 0' the momentum is the same in magnitude as before, but its
**vector**is displaced from OF to O'F'. - The moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the
**vector**of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o. - This is the simplest case of generation of a plane figure by a moving ordinate; the corresponding figure for generation by rotation of a radius
**vector**is a circle. **Vector**Analysis >>- Thus the path of the ray when the aether is at rest is the curve which makes fds/V least; but when it is in motion it is the curve which makes fds/(V+lug-m y -I-nw) least, where (l,m,n) is the direction
**vector**of Ss. - In octonions the analogue of Hamilton's
**vector**is localized to the extent of being confined to an indefinitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to p, and, like Hamilton's**vector**, wp is not localized; wp is therefore called a**vector**, though it differs from Hamilton's**vector**in that the product of any two suchwp and coo- is zero because w 2 =o.**vectors** - The plane is of
**vector**magnitude ZVq, its equation is ZSpq=Sr, and its expression is the bi-quaternion nVq+wSr; the point is of scalar magnitude 4Sq, and its position**vector**is [3, where 1Vf3q=Vr (or what is the same, fi = [Vr+q. - P is a rotor and coo- a
**vector**), is called a motor, and has the geometrical significance of Ball's wrench upon, or twist about, a screw. - As 6t is indefinitely diminished, the
**vector**OU will tend to a definite limit OV; this is adopted as the definitiov of the velocity of the moving point at the instant t. - For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS;
**VECTOR**ANALYSIS. - Kuhn (1750-1751) and Jean Robert Argand (1806) were completed by Karl Friedrich Gauss, and the formulation of various systems of
**vector**analysis by Sir William Rowan Hamilton, Hermann Grassmann and others, followed. - Hence th,l moment of the momentum (considered as a localized
**vector**) about 0 will be constant.