. Differential and integral calculus, an introductory course for colleges and engineering schools. yQ = r cos /3 J»»*c / Kt A \r x-x\ X / mr, z — Zq = r cos 7 = nr, 282 GEOMETRY OF THREE DIMENSIONS §190 whence (III) x = x0 + Ir, y = yoJr mr, z = Zq-\- nr. Now if r be variable, so also are x, y, and z, and as r varies from— oo to +oo, the point (x, y, z) traces the line. Equations (III)are termed the parametric equations of the line, r being the variableparameter. 190. The Normal Equation of the Plane. Let p be the lengthand a, (3, y the direction angles of the perpendicular (the normal),
. Differential and integral calculus, an introductory course for colleges and engineering schools. yQ = r cos /3 J»»*c / Kt A \r x-x\ X / mr, z — Zq = r cos 7 = nr, 282 GEOMETRY OF THREE DIMENSIONS §190 whence (III) x = x0 + Ir, y = yoJr mr, z = Zq-\- nr. Now if r be variable, so also are x, y, and z, and as r varies from— oo to +oo, the point (x, y, z) traces the line. Equations (III)are termed the parametric equations of the line, r being the variableparameter. 190. The Normal Equation of the Plane. Let p be the lengthand a, (3, y the direction angles of the perpendicular (the normal), OQ, let fall from the originupon a given plane. Let Pwith the coordinates x, y, z beany point of the plane. Nowpis the projection uponOQ ofur-X the line OP. Hence by B or(c) of Art. 187 we have x cos a + y cos /3 + z cos y = por (IV) Ix -j- my + nz = p and this is the normal equation of the plane. The equation Ax + By + Cz + D = 0 can be thrown into the normal form by dividing by the squareroot of the sum of the squares of the coefficients of x, y, and , if R = A2 + B2 + C2, then. (V) A , B , C D —==. x + —-7= y H 7= 2 = 7= Vfl VRy Vr Vr is the normal equation of the plane. Comparing (V) with (IV), we see that to the plane, and A, JL, JL Vr Vr Vr -D are the direction cosines of the normal Vr is the length of this normal, the radical being given the sign contrary to that of D, so that -D Vr shall be §191 THE RIGHT LINE IN SPACE 283 positive. From (V) it follows that in the equation of any plane thecoefficients of the variables are proportional to the direction cosinesof the normal to the plane. The equations of a line through the point (xq, Vq, z0) normal(perpendicular) to the plane Ax + By + Cz + D = 0 are there-fore(VI*) x = x0 + —,= r, y = 2/o + —?= r, z = z0 + —== r, where r is the variable parameter. fNow we may write r in place of —=., and the equations of the VR normal line take the simpler form (VI) x = Xq + Ar, y = y0-\- Br, z = z0 + ang
Size: 1588px × 1574px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
