vector analysis - Encyclopædia Britannica

7/29/2019 vector analysis - Encyclopdia Britannica

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vector analysis

Encyclopdia Britannica

vector analysis, a branch of mathematics that deals with quantities that have both

magnitude and direction. Some physical and geometric quantities, called scalars, can be

fully defined by specifying their magnitude in suitable units of measure. Thus, mass can

be expressed in grams, temperature in degrees on some scale, and time in seconds.

Scalars can be represented graphically by points on some numerical scale such as a clock

or thermometer. There also are quantities, called vectors, that require the specification of

direction as well as magnitude. Velocity, force, and displacement are examples of

vectors. A vector[1] quantity can be represented graphically by a directed line segment,

symbolized by an arrow pointing in the direction of the vector quantity, with the length

of the segment representing the magnitude of the vector.

Vector algebra.

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 vectorAB 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.

The equality of two vectors a and b is denoted by the usual symbolic notation a = b, and

useful definitions of the elementary algebraic operations on vectors are suggested by

geometry. Thus, ifAB = a in Figure 1 represents a displacement of a particle from A to Band subsequently the particle is moved to a position C, so that BC= b, it is clear that the

displacement from A to Ccan be accomplished by a single displacement AC= c. Thus, it

is logical to write a + b = c. This construction of the sum, c, ofa and b yields the same

result as the parallelogram law in which the resultant c is given by the diagonal ACof the

parallelogram constructed on vectors AB andAD as sides. Since the location of the initial

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point B of the vectorBC= b is immaterial, it follows that BC= AD. Figure 1 shows that

AD + DC= AC, so that the commutative law [2]

holds for vector addition. Also, it is easy to show that the associative law [3]

is valid, and hence the parentheses in (2) can be omitted without any ambiguities.

Ifs is a scalar, sa or as is defined to be a vector whose length is |s||a| and whose direction

is that ofa when s is positive and opposite to that ofa ifs 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 (1), (2), and (3) are identical with those encountered in ordinary

algebra, it is quite proper to use familiar algebraic rules to solve systems of linear

equations [4] containing vectors. This fact makes it possible to deduce by purely

algebraic means many theorems of synthetic Euclidean geometry[5]

that require

complicated geometric constructions.

Products of vectors.

The multiplication ofvectors leads to two types of products, the dot product [6] and the

cross product[7]

.

The dot or scalar product of two vectors a and b, written ab, is a real number [8] |a||b|

cos (a,b), where (a,b) denotes the angle between the directions ofa and b. Geometrically,

Ifa and b are at right angles then ab = 0, and if neither a nor b is a zero vector then the


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vanishing of the dot product shows the vectors to be perpendicular. Ifa = b then cos (a,b)

= 1, and aa = |a|2 gives the square of the length ofa.

The associative, commutative, and distributive laws [9] of elementary algebra [10] are

valid for the dot multiplication ofvectors.

The cross or vector product of two vectors a and b, written ab, is the vector

where n is a vector of unit length perpendicular to the plane ofa and b and so directed

that a right-handed screw rotated from a toward b will advance in the direction ofn (see

Figure 2). Ifa and b are parallel, ab = 0. The magnitude ofab can be represented by

the area of the parallelogram having a and b as adjacent sides. Also, since rotation from b

to a is opposite to that from a to b,

This shows that the cross product is not commutative, but the associative law (sa) b =

s(ab) and the distributive law

are valid for cross products.

Coordinate systems.

Since empirical laws [11] of physics [12] 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 [13] or coordinate system [14] 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 numbersthat 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 vectorA can be expressed uniquely as the diagonal of the parallelepiped whose

edges are the components ofA in the directions of the base vectors. In common use is a


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set of three mutually orthogonal unit vectors (i.e.,vectors of length 1) i, j, kdirected

along the axes of the familiar Cartesian reference frame (see Figure 3). In this system the

expression takes the form

where x, y, and z are the projections ofA upon the coordinate axes. When two vectors A1andA2 are represented as

then the use of laws (3) yields for their sum

Thus, in a Cartesian frame, the sum ofA1 and A2 is the vector determined by (x1 + y1,x2+ y2, x3 +y3). 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 ofi, j, and kin (9).

Ifvectors are represented by 1 3 (or 3 1) matrices consisting of the components

(x1,x2, x3) of the vectors, it is possible to rephrase formulas (7) through (9) in the

language of matrices. 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


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reference frame determined by the set of base vectors a1,a2,a3,a4 with components

determined by the rows of the matrix

A vectorx is then represented in the form

so that in a four-dimensional space, every vector is determined by the quadruple of the

components (x1,x2,x3,x4).

Calculus of vectors.A particle moving in three-dimensional space can be located at each instant of time tby a

position vector[15]r drawn from some fixed reference point O. Since the position of the

terminal point ofr depends on time, r is a vector function oft. Its components in the

directions of Cartesian axes, introduced at O, are the coefficients ofi, j, and kin the

representation

If these components are differentiable functions [16], the derivative [17] ofr with respect

to tis defined by the formula

which represents the velocity v of the particle. The Cartesian components ofv appear as

coefficients ofi, j, and kin (10). If these components are also differentiable, the

acceleration a = dv/dtis obtained by differentiating (10):

The rules for differentiating products of scalar functions remain valid for derivatives of

the dot and cross products ofvector functions, and suitable definitions of integrals of

vector functions allow the construction of the calculus [18] ofvectors, which has become


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a basic analytic tool in physical sciences and technology.

http://www.britannica.com/EBchecked/topic/1240588/vector1.

http://www.britannica.com/EBchecked/topic/129419/commutative-law2.

http://www.britannica.com/EBchecked/topic/39468/associative-law3.

http://www.britannica.com/EBchecked/topic/342131/linear-equation4.

http://www.britannica.com/EBchecked/topic/194901/Euclidean-geometry5.

http://www.britannica.com/EBchecked/topic/288565/inner-product6.

http://www.britannica.com/EBchecked/topic/624362/vector-product7.

http://www.britannica.com/EBchecked/topic/492990/real-number8.

http://www.britannica.com/EBchecked/topic/166204/distributive-law9.

http://www.britannica.com/EBchecked/topic/184192/elementary-algebra10.

http://www.britannica.com/EBchecked/topic/332743/law11.

http://www.britannica.com/EBchecked/topic/458757/physics12.

http://www.britannica.com/EBchecked/topic/495116/reference-frame13.

http://www.britannica.com/EBchecked/topic/136400/coordinate-system14.

http://www.britannica.com/EBchecked/topic/471784/position-vector15.

http://www.britannica.com/EBchecked/topic/162882/differentiable-function16.

http://www.britannica.com/EBchecked/topic/158518/derivative17.

http://www.britannica.com/EBchecked/topic/89161/calculus18.


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vector analysis - Encyclopædia Britannica

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