An elementary treatise on coordinate geometry of three dimensions . - 2, 2). Find the equationto the cone. Ans. 4x2 + 4yi + 4z- + 9y: + 9:j- + \\>y = 0. Ex. 7. Find the equation to the right circular cylinder of radius 2whose axis passes through (1, 2, 3) and has direction-cosines pro-portional to (2, - 3, (5). Ans. 9(2y + z - 7)2 + 4(s - 3.*:)2 + (3r + 2y - If = 196. *25. Direction-cosines referred to oblique axes. LetXOX, YOY, ZOZ, (tig. 20), be oblique axes, the anglesYOZ, ZOX, XOY being X, /m, v respectively. Let AOA be theline through O whose direction-cosines are cosoc, cos/3,cos y. T
An elementary treatise on coordinate geometry of three dimensions . - 2, 2). Find the equationto the cone. Ans. 4x2 + 4yi + 4z- + 9y: + 9:j- + \\>y = 0. Ex. 7. Find the equation to the right circular cylinder of radius 2whose axis passes through (1, 2, 3) and has direction-cosines pro-portional to (2, - 3, (5). Ans. 9(2y + z - 7)2 + 4(s - 3.*:)2 + (3r + 2y - If = 196. *25. Direction-cosines referred to oblique axes. LetXOX, YOY, ZOZ, (tig. 20), be oblique axes, the anglesYOZ, ZOX, XOY being X, /m, v respectively. Let AOA be theline through O whose direction-cosines are cosoc, cos/3,cos y. Take P, (x, y, z) any point on AOA. and let themeasure of OP be r. Draw PN parallel to OZ to meetthe plane XOY in N, and NM parallel to OY to meet OXin M. Then, since the projection of OP is equal to the 26 COORDINATE GEOMETRY [ch. ii. sum of the projections of OM, MN, NP, projecting on OX,OY, OZ, OP in turn, we obtain r cos a. = x + y cos v + z cos m, (1) r cos /3 = x cos v+y+z cos X, (2) r cos y = x cos /x + y cos X + 0, (3) r = #cos<x+2/cos/3 + 3COSy (4) (z. Fig. 20. Therefore, eliminating r, cc, y, 0, we have the relationsatisfied by the direction-cosines of any line 1, cos v, cos ^ cos a =0,cosj/, 1, cosX, cos/3 cos fx, cos X, 1, cos y cosoc, cos/3, cosy, 1 which may be written, X sin2X cos2oc— 22(cos X — cos /x cos y)cos /3 cos y = 1 — cos2 X — cos2 [X — cos2 v+2 cos X cos ya cos j/. Cor. 1. Multiply (1), (2), (3) by x, y, z respectively, andadd, then x2 + y2 + z2 + 2yz cos X + 2zx cos /x + Ixy cos 1/= r (x cos oc +1/ cos /3 + z cos y),= r2, [by (4)] (A) Cor. 2. If P, Q are (x1? yx, zx), (x2, i/2, 02), PQ2 is given by2(x2 - x1f+2X(y2 - yx)(z2 - %)cos X. §§25-27] THE ANGLE BETWEEN TWO LINES 27 If P, (.,//, .) is any point on the plane through O at rightangles to OX, the projection of OP on OX u zero, and then ./•+//ci >s v + I C08 /. —0. Ex. 2. If P, (.v, y, z) is any point on the normal through O to theplane XOY, x+y cos v + .jcos/a = 0 = .>c
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