By Luther Pfahler Eisenhart

This textbook explores the configurations of issues, traces, and planes in area outlined geometrically, interprets them into algebraic shape utilizing the coordinates of a consultant element of the locus, and derives the equations of the conic sections. The Dover version is an unabridged republication of the paintings initially released by way of Ginn and corporate in 1939.

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Example text

In fact, in Fig. 6 the sensed segments PiQ and QP of the line through the points , 5, give the direction numbers of PiP, and PiQ 2 and Q 2 P 2 the direction numbers of P\Pz. 1) is the algebraic statement of the characteristic property of a line that any two segments having an end point in common have the same direction. If we say that direction numbers of any segment of a line are direction numbers of the line, it follows that a line has an endless number of sets of direction numbers, but that the numbers of any set are proportional to those of any other set.

Show 7. that the coordinates of any point on a line through the numbers of the line. origin are direction Show 8. 5) hold for a line parallel to either if we make the convention that "the angle between" coordinate axis the positive directions of two parallel lines 9. Show zero. is that for the lines in Figs. 3) A + cos 2 A = l. is 2 equivalent to the trigonometric identity sin 10. Show that (x ~ * i)( * 2 - *3) + (y ~ y^* - *) = o two lines through the point (xi, y\) parallel and perpendicular respectively to the line through the points (# 2 yz) and are equations of , (*s, y*)- Given the 11.

For what value of a is the line segment from the point P\(2 to the point P(3, a) perpendicular to the line is line 2). Find the coordinates of the point p(x, y) so that the line ment P\P has the same direction and is twice as long as the 5. Pa 1) (3, 3). segment joining Pi(l, 6. and 1) is isosceles. vertices of 4. that the triangle with the vertices the point (5, 1) ? 16 y 1) segment PiP2 where , Internal Sec. 4] 7. and External Division of a Line Segment Find the lengths and direction cosines of the sides of the whose vertices are (3, also the 6), (8, 2), and ( 1, 1) triangle ; cosines of the angles of the triangle.

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