An elementary treatise on coordinate geometry of three dimensions . D,, B,/Dlt Cj/Di, IA2/D,, etc. i A,, B„ C,,A.,, etc. (,-A3 ; (C. Smith, Algebra, p. 644.) *Ex. 9. The lengths of the edges OA, OB, OC of a tetrahedronOABC are a, b, c, and the angles BOC, COA, AOB are A, /x, v ; find the volume. Suppose that the direction-cosines of OA, OB, OC, referred trectangular axes through O, are lly m,, ??, ; l2, «?.._,, n2; £3, ni3, n3 ; thenthe coordinates of A are lxa, mxa, nxa, etc. Therefore 6. Vol. OABC= ]lxa, m^a, nxa\ I J>, m.,b, n2bI Lp, m3c, n3e \ = ±abc ^2, s& 2*1*8 - = ± a
An elementary treatise on coordinate geometry of three dimensions . D,, B,/Dlt Cj/Di, IA2/D,, etc. i A,, B„ C,,A.,, etc. (,-A3 ; (C. Smith, Algebra, p. 644.) *Ex. 9. The lengths of the edges OA, OB, OC of a tetrahedronOABC are a, b, c, and the angles BOC, COA, AOB are A, /x, v ; find the volume. Suppose that the direction-cosines of OA, OB, OC, referred trectangular axes through O, are lly m,, ??, ; l2, «?.._,, n2; £3, ni3, n3 ; thenthe coordinates of A are lxa, mxa, nxa, etc. Therefore 6. Vol. OABC= ]lxa, m^a, nxa\ I J>, m.,b, n2bI Lp, m3c, n3e \ = ±abc ^2, s& 2*1*8 - = ± abc 1, COS I, COS//. -. 17,/,, 2J22, 2% COS 1, 1, cos A zhh> yz1 2*32 COS it, cos A, 1 (Cf. § 27, Ex. a) [CH. IV. CHAPTER IV. CHANGE OF AXES. 52. OX, OY, OZ; o£ Or], o£ are two sets of rectangularaxes through a common origin O, and the direction-cosinesof o£ Oi], o£, referred to OX, OY, OZ, are lx, m1? n^, l2, w2, ;l3, m3, ?i3. P, any point, has coordinates x, y, z referred toOX, OY, OZ and £ q, £ referred to 0£ Orj, o£. We have to. Pig. 26. express x, y, z in terms of £, r], £ and the direction-cosines,and vice-versa. In the accompanying figure, ON, NM, MP represent £ rj, £,and OK, KL, LP represent x, y, z. Projecting OP and ON,NM, MP on OX, OY, OZ in turn, we obtain X= l^+ 1& + m y=m1£+m2r,+m3g,V .... (1) §§52,53] THREE PERPENDICULAR MM;.- And projecting op and ok. kl, lp on ot,:. o>/, oc in turnwe obtain / *2 ///. l f *S .; :; The equations (1) and (2) can be derived from theabove scheme, which may be constructed as follows: Affixto the columns and rows the numbers x, y, z; g, t], £: andin the square common to the column headed x and therow headed £, place the cosine of the angle between OXand Og, lx, and so on. To obtain the value of x, multiplythe numbers in the ^-column by the numbers at the leftof their respective rows and add the products: to obtainthe value of g, multiply the numbers in the £-row by the
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
