Relations Connecting Coordinates 6. The Distance Between Two Points 7. Defining Figures 8. We Begin to Solve Problems 9. Other Systems of Coordinates Chapter 3. The Coordinates of a Point in Space Coordinate Axes and Planes Defining Figures in Space Part 2. Introduction 1. Some General Considerations 2. Geometry as an Aid in Calculation 3. The Peculiarities of Four-Dimensional Space 5.
Some Physics Chapter 2. Four-Dimensional Space 6. Coordinate Axes and Planes 7. Some Problems Chapter 3. You can change your ad preferences anytime. Upcoming SlideShare. Like this document? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Published in: Education. Full Name Comment goes here. Accordingly the positive sense along a line is upward, that is, y increasing, when the line is not parallel to the x-axis; and to the right, that is, x increasing, when the line Since all segments of a line have the is parallel to the x-axis.
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Plane we have corollary of the theorem Two [6. Hence we have An [6. We seek next an equation of any line perpendicular to the with equation 6. Theorem [6. What are the direction cosines of the line 3 x of a line perpendicular to it? Find the angle between the positive directions of the perpendiculars to each of the following pairs of lines : a.
Find the angles between the positive directions of the 6.
Show zero. Does this prove that these C c, 0 ; ; results are true for When the line is parallel to the j-axis, the denominator in the right-hand member of 7. Hence we say that the slope is not defined for a line parallel to the jy-axis some writers ; say that in this case the slope is infinite. For a given value of m equation 7.
An equation lines, of any A. We have introduced the concept of slope because of its and particularly because of its use in the application of the differential calculus to the study of lines and curves in the plane, an interesting subject awaiting the reader. On the other hand, we have emphasized direction numbers and direction cosines because they may be applied to any line withtraditional use, out exception, and, as we shall see in Chapter 2, are applicable to the study of lines in space, whereas the concept of slope is not used in this connection.
Directed Distance from a Line to a Point Sec.
Find equations of the two lines through the point 1, 3 forming with the x-axis an isosceles triangle with the point as vertex and the base angles 30 For what value of 5. How 8. Find equations of the make is Theorem angles of 45 a consequence of Ex.
What are the a pair of straight slopes of these lines? Draw the graph of a line and indicate the various quantities appearing in equations 6. One might question whether perhaps this value has something to do with the distance from the line to the point. In tion that the direction upward along a line inclined to the axes is positive, and showed that the direction cosine JJL is always positive, and that X is positive or negative according as inclined as in Fig. In consequence of this result, and of Theorems [6.
We denote by d the distance from P 2 to PI, d being a positive number, since it is in the positive direction measured of the line P2Pi. PI lies well provided We consider the other case the line 8. Comparing the line PiP 2 in Fig. Com- paring these equations with 8. Con- sequently equations 8.
The reader may verify that equations 8. Consider now the equation 8 6. Since the equation is of the first degree and y, it is an equation of a line. Since this is true of every point on the line 8. Find equations of the two at the distance 2 from this line. How line are numerically equal to 3. Show that p is the directed distance from the line to the origin.
In accordance 1, each of equations 9. When the left-hand member of either equation is a constant multiple of the left-hand member of the other, the two equations have 1 that is, they are not the same solutions, as remarked in ; 39 Points and Lines in the Plane [Chap. But when the equations are essentially different, x and y have, in general, different mean- two equations. To emphasize this fact, different might more appropriately be used in place of x and y ings in the letters but this is not the general practice; to the customary practice of using the same letters in both equations and expect the reader to bear the in the second equation, we adhere x and y distinction in mind.
We observe that the second determinants in 9. We first consider equations 9. Moreover, since any common solution of equations 9. We call the above determinant, that is, the determinant of the coefficients of x and y in equations 9. Accordingly we have [9. Accordingly the geometric the following: [9.
This means that the assumption a common solution of equations 9.
Hence we have the theorem 9. When this value for 02 is substituted in 9. From 9. Furthermore, when 9. Cl 02 C2 Accordingly is not equal to zero, we we have equations 9. Two mon : equations of the first degree in x and y have no comsolution if to within possible constant factors they are of the forms 9. In accordance with Theorem [6.
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By Theorem [9. The reader may have been told that two parallel lines meet in a point at infinity.
Dedekind’s Contributions to the Foundations of Mathematics
A point at infinity so defined is not like the it is a concept sometimes points with which we are dealing he may make the general so that introduced by the geometer statement that any two noncoincident lines meet in a point. This does not mean that any value of tions 9. We shall show that it means that either equation is a constant multiple of the other. We have seen that equation 9. All three of the determinants 9.