# cal1 - Chapter One Euclidean Three-Space 1.1 Introduction....

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1.1 Chapter One Euclidean Three-Space 1.1 Introduction. Let us briefly review the way in which we established a correspondence between the real numbers and the points on a line, and between ordered pairs of real numbers and the points in a plane. First, the line. We choose a point on a line and call it the origin . We choose one direction from the origin and call it the positive direction. The opposite direction, not surprisingly, is called the negative direction. In a picture, we generally indicate the positive direction with an arrow or a plus sign: Now we associate with each real number r a point on the line. First choose some unit of measurement on the line. For r 0, associate with r the point on the line that is a distance r units from the origin in the positive direction. For r < 0, associate with r the point on the line that is a distance r units from the origin in the negative direction. The number 0 is associated with the origin. A moments reflection should convince you that this procedure establishes a so-called one-to-one correspondence between the real numbers and the points on a line. In other words, a real number determines exactly one point on a line, and, conversely, a point on the line determines exactly one real number. This line is called a real line . Next we establish a one-to-one correspondence between ordered pairs of real numbers and points in a plane. Take a real line, called the first axis , and construct another real line, called the second axis , perpendicular to it and passing through the origin of the first axis. Choose this point as the origin for the second axis. Now suppose we have an ordered pair ( , ) x x 1 2 of reals. The point in the plane associated with this ordered pair is

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1.2 found by constructing a line parallel to the second axis through the point on the first axis corresponding to the real number x 1 , and constructing a line parallel to the first axis through the point on the second axis corresponding to the real number x 2 . The point at which these two lines intersect is the point associated with the ordered pair ( , ) x x 1 2 . A moments reflection here will convince you that there is exactly one point in the plane thus associated with an ordered pair ( a , b ), and each point in the plane is the point associated with some ordered pair ( a , b ): It is traditional to assume the point of view we have taken in this picture, in which the first axis is horizontal, the second axis is vertical, the positive direction on the first axis is to the right, and the positive direction on the second axis is up. We thus usually speak of the horizontal axis and the vertical axis, rather than the first axis and the second axis. We also frequently abuse the language by speaking of a point ( , ) x x 1 2 when, of course, we actually mean the point associated with the ordered pair ( , ) x x 1 2 . The numbers x 1 and x 2 are called the coordinates of the point- x 1 is the first coordinate and x 2 is the second coordinate. Given any collection of ordered pairs( A collection of ordered pairs is called a
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## This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal1 - Chapter One Euclidean Three-Space 1.1 Introduction....

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