An elementary treatise on coordinate geometry of three dimensions . 7M, = ??1 = 0: l., = n2 = 0, \ ; l, — m?t = 0, lx = »?.>??3 — »?.,».,, ancoincides with OY [ 1 /C-i \M j M /CO \.) //lo 1 j [ ? //C~ V*, /(« 1 . d if 0£ be rotated to coincide with OX. O 72 COORDINATE GEOMETRY [CH. IV. 54. Section of a surface by a given plane. The follow-ing method of transformation can be applied with advantagewhen the section of a given surface by a given planepassing through the origin is to be considered. Let the equation to the plane be lx + 7)iy + nz = 0, wherel2+m2+n2=l, and n is positive. Take
An elementary treatise on coordinate geometry of three dimensions . 7M, = ??1 = 0: l., = n2 = 0, \ ; l, — m?t = 0, lx = »?.>??3 — »?.,».,, ancoincides with OY [ 1 /C-i \M j M /CO \.) //lo 1 j [ ? //C~ V*, /(« 1 . d if 0£ be rotated to coincide with OX. O 72 COORDINATE GEOMETRY [CH. IV. 54. Section of a surface by a given plane. The follow-ing method of transformation can be applied with advantagewhen the section of a given surface by a given planepassing through the origin is to be considered. Let the equation to the plane be lx + 7)iy + nz = 0, wherel2+m2+n2=l, and n is positive. Take as o£, the new axis of z, the normal to the planewhich passes through O and makes an acute angle withOZ. Then the equations to o£, referred to OX, OY, OZ, arex/l=yjm=z(n. Take as Orj, the new i/-axis, the line inthe plane ZO£ which is at right angles to o£ and makes anacute angle with OZ. Then choose Of, the new a>axis, atright angles to Ot] and o£, and so that the system o£ 0>j, 0£can be brought to coincidence with OX, OY, OZ. The given. Fig. 2S. plane is £0*7, and since O^ is at right angles to o£ and O)?,it is at right angles to OZ which lies in the plane £o>/.Hence o£ lies in the plane XOY, and therefore is the lineof intersection of the given plane and the plane XOY. Theequation to the plane gOrj is x/l = y/m; therefore if X, /a, vare the direction-cosines of O^, l\-\-m/i+nv=0,mX — Ifi = 0, X u v ±1 I whence — = ^- = ?m Z2 + m2 V^ + m2 g54] A USEFUL Ti:.\.\sioi:.\i.\l ION But o>] makes an acute angle with oz and then fonpositive, and therefore the negative sign must b taken inthe ambiguity. X = ?In — in n /* = > = >/P + m?. J¥+wCi JP+m*And since ot is at right angles to o>; hh<1 o£ by § 53 (E;.i he direction-cosines of o^ are // // — mv, lv — n X, niX — f/i; — ra ^ j i . i Hence we have the scheme: 0. * y , £ — 77) i 0 <x/£2 + rrb2 s/- + m2 — 1 n — mn s/-+///- »7 Jl-+ iir f I m n Ex. 1. Shew th
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
