vector

—vectorial /vek tawr"ee euhl, -tohr"-/, adj. —vectorially, adv.

/vek"teuhr/, n.

1. Math.

a. a quantity possessing both magnitude and direction, represented by an arrow the direction of which indicates the direction of the quantity and the length of which is proportional to the magnitude. Cf. scalar (def. 4).

b. such a quantity with the additional requirement that such quantities obey the parallelogram law of addition.

c. such a quantity with the additional requirement that such quantities are to transform in a particular way under changes of the coordinate system.

d. any generalization of the above quantities.

2. the direction or course followed by an airplane, missile, or the like.

3. Biol.

a. an insect or other organism that transmits a pathogenic fungus, virus, bacterium, etc.

b. any agent that acts as a carrier or transporter, as a virus or plasmid that conveys a genetically engineered DNA segment into a host cell.

4. Computers. an array of data ordered such that individual items can be located with a single index or subscript.

v.t.

5. Aeron. to guide (an aircraft) in flight by issuing appropriate headings.

6. Aerospace. to change direction of (the thrust of a jet or rocket engine) in order to steer the craft.

[1695-1705; < L: one that conveys, equiv. to vec-, var. s. of vehere to carry + -tor -TOR]

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In mathematics, a quantity characterized by magnitude and direction.

Some physical and geometric quantities, called scalars, can be fully defined by a single number specifying their magnitude in suitable units of measure (e.g., mass in grams, temperature in degrees, time in seconds). Quantities like velocity, force, and displacement must be specified by a magnitude and a direction. These are vectors. A vector quantity can be visualized as an arrow drawn in a specific direction, whose length is equal to the magnitude of the quantity represented. A two-dimensional vector is specified by two coordinates, a three-dimensional vector by three coordinates, and so on. Vector analysis is a branch of mathematics that explores the utility of this type of representation and defines the ways such quantities may be combined. See also vector operations.

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▪ mathematics

      in mathematics, a quantity that has both magnitude and direction but not position. Examples of such quantities are velocity and acceleration. In their modern form, vectors appeared late in the 19th century when Josiah Willard Gibbs (Gibbs, J Willard) and Oliver Heaviside (Heaviside, Oliver) (of the United States and Britain, respectively) independently developed vector analysis to express the new laws of electromagnetism discovered by the Scottish physicist James Clerk Maxwell (Maxwell, James Clerk). Since that time, vectors have become essential in physics, mechanics, electrical engineering, and other sciences to describe forces mathematically.

      Vectors may be visualized as directed line segments whose lengths are their magnitudes. Since only the magnitude and direction of a vector matter, any directed segment may be replaced by one of the same length and direction but beginning at another point, such as the origin of a coordinate system. Vectors are usually indicated by a boldface letter, such as v. A vector's magnitude, or length, is indicated by |v|, or v, which represents a one-dimensional quantity (such as an ordinary number) known as a scalar. Multiplying a vector by a scalar changes the vector's length but not its direction, except that multiplying by a negative number will reverse the direction of the vector's arrow. For example, multiplying a vector by 1/2 will result in a vector half as long in the same direction, while multiplying a vector by −2 will result in a vector twice as long but pointed in the opposite direction.

 Two vectors can be added or subtracted. For example, to add or subtract vectors v and w graphically (see the diagram—>), move each to the origin and complete the parallelogram formed by the two vectors; v + w is then one diagonal vector of the parallelogram, and v − w is the other diagonal vector.

 There are two different ways of multiplying two vectors together. The cross, or vector, product results in another vector that is denoted by v × w. The cross product magnitude is given by |v × w| = vw sin θ, where θ is the smaller angle between the vectors (with their “tails” placed together). The direction of v × w is perpendicular to both v and w, and its direction can be visualized with the right-hand rule, as shown in the figure—>. The cross product is frequently used to obtain a “normal” (a line perpendicular) to a surface at some point, and it occurs in the calculation of torque and the magnetic force on a moving charged particle.

      The other way of multiplying two vectors together is called a dot product, or sometimes a scalar product because it results in a scalar. The dot product is given by v ∙ w = vw cos θ, where θ is the smaller angle between the vectors. The dot product is used to find the angle between two vectors. (Note that the dot product is zero when the vectors are perpendicular.) A typical physical application is to find the work W performed by a constant force F acting on a moving object d; the work is given by W = Fd cos θ.

▪ physics

      in physics, a quantity that has both magnitude and direction. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity's magnitude. Although a vector has magnitude and direction, it does not have position. That is, as long as its length is not changed, a vector is not altered if it is displaced parallel to itself.

      In contrast to vectors, ordinary quantities that have a magnitude but not a direction are called scalars (scalar). For example, displacement, velocity, and acceleration are vector quantities, while speed (the magnitude of velocity), time, and mass are scalars.

      To qualify as a vector, a quantity having magnitude and direction must also obey certain rules of combination. One of these is vector addition, written symbolically as A + B = C (vectors are conventionally written as boldface letters). Geometrically, the vector sum can be visualized by placing the tail of vector B at the head of vector A and drawing vector C—starting from the tail of A and ending at the head of B—so that it completes the triangle. If A, B, and C are vectors, it must be possible to perform the same operation and achieve the same result (C) in reverse order, B + A = C. Quantities such as displacement and velocity have this property ( commutative law), but there are quantities (e.g., finite rotations in space) that do not and therefore are not vectors.

      The other rules of vector manipulation are subtraction, multiplication by a scalar, scalar multiplication (also known as the dot product or inner product), vector multiplication (also known as the cross product), and differentiation. There is no operation that corresponds to dividing by a vector. See vector analysis for a description of all of these rules.

      Although vectors are mathematically simple and extremely useful in discussing physics, they were not developed in their modern form until late in the 19th century, when Josiah Willard Gibbs (Gibbs, J Willard) and Oliver Heaviside (Heaviside, Oliver) (of the United States and England, respectively) each applied vector analysis in order to help express the new laws of electromagnetism, proposed by James Clerk Maxwell (Maxwell, James Clerk).

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