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COMPONENTS OF A VECTOR Any vector A in 3 dimensions can be represented with initial point at the origin O of a rectangular coordinate system [see Fig. 7-7]. Let A1 ; A2 ; A3 be the rectangular coordinates of the terminal point of vector A with initial point at O. The vectors A1 i; A2 j; and A3 k are called the rectangular component vectors, or simply component vectors, of A in the x, y; and z directions respectively. A1 ; A2 ; and A3 are called the rectangular components, or simply components, of A in the x, y; and z directions respectively. The vectors of the set fi; j; kg are perpendicular to one another, and they are unit vectors. The words orthogonal and normal, respectively, are used to describe these characFig. 7-7 teristics; hence, the set is what is called an orthonormal basis. It is easily shown to be linearly independent. In an n-dimensional space, any set of n linearly independent vectors is a basis. The further characteristic of a basis is that any vector of the space can be expressed through it. It is the basis representation that provides the link between the geometric and algebraic expressions of vectors and vector concepts. The sum or resultant of A1 i; A2 j; and A3 k is the vector A, so that we can write A A1 i A2 j A3 k The magnitude of A is A jAj q A2 A2 A2 1 2 3 2 1

Default domain, zone and authority Domain root cache prime information Local domain information Reverse query information

The appropriate command can then be accessed. WorldPlot[World]

Once a package is loaded you can get a list of the functions it contains by using the Names command.

EXAMPLE 27 (Continuation of Example 26)

In particular, the position vector or radius vector r from O to the point x; y; z is written p and has magnitude r jrj x2 y2 z2 : r xi yj zk 3

Names["WorldPlot`*"] {Africa, Albers, Asia, ContiguousUSStates, Equirectangular, Europe, LambertAzimuthal, LambertCylindrical, Mercator, MiddleEast, Mollweide, NorthAmerica, Oceania, Orthographic, RandomColors, RandomGrays, ShowTooltips, Simple, Sinusoidal, SouthAmerica, ToMinutes, USData, USStates, World, WorldBackground, WorldBorders, WorldClipping, WorldCountries, WorldData, WorldDatabase, WorldFrame, WorldFrameParts, WorldGraphics, WorldGrid, WorldGridBehind, WorldGridStyle, WorldPlot, WorldPoints, WorldProjection, WorldRange, WorldRotatedRange, WorldRotation, WorldToGraphics}

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All the machines in the organization which will be using DNS need to know where to find the primary and secondary name servers. This information is recorded in a resolv.conf file on each machine (Example 2.3). When a domain name is resolved to an IP address, a query is sent to the first name server in the resolv.conf list. If timeout period expires before an answer is received from the first name server, then the second name server is queried.

Calendar` Names["Calendar`*"] {Calendar, CalendarChange, DateQ, DayOfWeek, DaysBetween, DaysPlus, EasterSunday, EasterSundayGreekOrthodox, Friday, Gregorian, Islamic, Jewish, JewishNewYear, Julian, Monday, Saturday, Sunday, Thursday, Tuesday, Wednesday} DaysBetween

DOT OR SCALAR PRODUCT The dot or scalar product of two vectors A and B, denoted by A B (read A dot B) is de ned as the product of the magnitudes of A and B and the cosine of the angle between them. In symbols, A B AB cos ; 0@ @ 4

DaysBetween[{year1, month1, day1}, {year2, month2, day2}] gives the number of days between the dates {year1, month1, day1} and {year2, month2, day2}. DaysBetween[{year1, month1, day1, hour1, minute1, second1}, {year2, month2, day2, hour2, minute2, second2}] gives the number of days between the given dates.

DaysBetween[{2007,8,3},{2008,12,5}] 490

1.35 The function DayOfWeek appears in the package Calendar` and gives the day of the week of any date in the calendar. Load the package, obtain help to determine its syntax, and then determine which day of the week January 1, 2000, was.

Assuming that neither A nor B is the zero vector, an immediate consequence of the de nition is that A B 0 if and only if A and B are perpendicular. Note that A B is a scalar and not a vector. The following laws are valid: 1. A B B A Commutative Law for Dot Products

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DayOfWeek[{year, month, day}] gives the day of the week on which the given date {year, month, day} occurred. DayOfWeek[{year, month, day, hour, minute, second}] gives the day of theweek for the given date.

1.36 The package Combinatorica` contains functions in combinatorics and graph theory. One of these is KSubsets, which lists all subsets of size k of a given set. Load the package and execute Ksubsets[{1,2,3,4,5},3].

Combinatorica` KSubsets[{1, 2, 3, 4, 5}, 3] {{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}

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