Vector Subtraction Research Articles

Domino Rule. A Way of Handling Relative Velocities

Relative velocities can be combined in a way similar to the positioning of dominoes in the parlor game. The advantage of this method over the more usual one involving addition or subtraction of vectors is that it is more intuitive. It is particularly useful in the transition between laboratory and center of mass coordinate systems.

Nearshore ocean currents off San Diego, California

Observations of the trajectories of fields of free-drifting current drogues were made off San Diego, California, on 103 days. This provided data for statistical analysis of the surface current regime in two nearshore areas separated by the entrance to San Diego Bay. Reduction of the data required corrections for wind drag on the exposed portion of the drogues and for surface current drag on the float system of the subsurface drogues. Corrected subsurface data and surface data at times of low wind speeds were used to determine tidal components. Vector subtraction of tidal currents from total surface currents provided the components of the wind-induced currents. These were then statistically adjusted as derived from the yearly wind distribution, and finally the tidal and wind-induced components were recombined to yield a statistical redistribution of the total surface current.

Characterizations of convex sets by local support properties

It is our purpose to establish some new characterizations of convex sets by means of local properties and to derive as a consequence certain known results. This will be done for sets in a topological linear space L, such a space being a real linear space with a Hausdorif topology such that the operations of vector addition x+y and scalar multiplication aex are continuous in both variables jointly [3]. The principal results are contained in Theorems 4 and 5. In order to describe matters simply, the following notations are used. NOTATIONS. The interior, closure, boundary and convex hull of a set S in L are denoted by int S, cl S, bd S and conv S respectively. The closed line segment joining xCS and yCS is indicated by xy, whereas L(x, y) stands for the line determined by x and y. The interior of a set S relative to the minimal linear variety containing it is denoted by intv S. Set union, intersection and difference are denoted by Y, . and respectively. Vector addition and subtraction are denoted by + and respectively. We let 0 and 4 stand for the empty set and the origin of L respectively. In the statements of theorems and definitions the names of previous authors are indicated for historical purposes. DEFINITION 1. Let SCL. A point xEbd S is called a point of mild convexity of S if x is not the midpoint of any segment uv with 0 uv f-'x Cint S. It is desirable to compare this definition with those given by Tietze [5] and by Leja and Wilkosz [4]. See also Kaufman [2]. For a brief summary of earlier results see Bonnesen and Fenchel [1, p. 7]. DEFINITION 2. Let xEbd S, where SCL. The point x is a point of weak or strong convexity of S, or a point of weak or strong concavity of S, if there exists a neighborhood N(x) of x and a linear functional f with f(x) c such that the following conditions hold: (a) (Tietze). The point x is a point of weak convexity of S if f (y) > c with y C N(x) -x implies y i S. (For strong convexity replace f (y) > c by f(y) _c.) (b) (Leja and Wilkosz). The point x is a point of strong concavity of S if f(y) < c with yCN(x) -x implies yES. (For weak concavity replace f(y) ?c by f(y) <c.)

Graphical Treatment of the Rotating Field

The object of this paper is to evolve diagrams by means of which nearly all the phenomena of the rotating field may be easily studied, and the various factors necessary for the calculation of fluxes, exciting current, etc., exactly determined. In order to avoid too frequent repetition of certain expressions, the author has adopted the expedient of using the words “addition” and “subtraction” to mean geometrical addition and subtraction of vectors according to the well-known conventional method.

The Elements of Quaternions (First Paper)

Extract HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.