. Plane and solid analytic geometry; an elementary textbook. oughly the nature of the surface : (a) x2 + y2 = 9, (d) y2 = ±z, (6) x-y + 2z = 10, (e) | + J + g = l, 3. What is the equation of the surface generated by therevolution of the hyperbola xy == k about the X-axis. 4. What is the position of a line whose equations arex + 3y = 10 and 3a- 4^ = 8? 5. The equations of any two surfaces may be representedby U=0 and V= 0, where U and Fare abbreviations foralgebraic expressions of any degree in x, y, and z. Showthat lU+kV=0 will represent a surface which passesthrough all the points common to t
. Plane and solid analytic geometry; an elementary textbook. oughly the nature of the surface : (a) x2 + y2 = 9, (d) y2 = ±z, (6) x-y + 2z = 10, (e) | + J + g = l, 3. What is the equation of the surface generated by therevolution of the hyperbola xy == k about the X-axis. 4. What is the position of a line whose equations arex + 3y = 10 and 3a- 4^ = 8? 5. The equations of any two surfaces may be representedby U=0 and V= 0, where U and Fare abbreviations foralgebraic expressions of any degree in x, y, and z. Showthat lU+kV=0 will represent a surface which passesthrough all the points common to the loci of U=0 andV=0, and which meets neither of these surfaces at anyother points. Show also that the locus of UV= 0 willconsist of the loci of U= 0 and V= 0. CHAPTER III THE PLANE 15. Normal form of the equation of a plane.—Let ON be the normal to the plane (a straight line of indefiniteextent perpendicular to the plane), and let «, /3, and 7 bethe angles which this normal makes with the axes. Letp be the perpendicular distance OK from the origin to. Fig. 11. the plane, measured along the normal. Let P(x,y,z)be any point in the plane. The line Pif will be perpen-dicular to ON, and the projection of OP on ON will beOK or p. But the projection of OP on ON is the sameas the projection on ON of the broken line OL, L C, CP, 215 216 ANALYTIC GEOMETRY OF SPACE [Ch. Ill, § 16 or x, y, z. From [5] the projection of OL on ON isx cos a; of LC y cos /3; of (7P, 2 cos 7. Hence a? cos a + ycosp + scos-y — p = 0. [1*^] This is called the normal form of the equation of aplane. The distance p is measured from the origin to theplane, and is positive or negative according as it runsin the positive or negative direction of the normal. Itis usually possible to choose the direction from the originto the plane as the positive direction of the normal,, sothat p will usually be a positive number. The angles a, /3, and 7 are measured from the positivedirections of the axes to the positive dire
Size: 1532px × 1631px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1900, bookpublishernewyo, bookyear1901
