. Plane and solid analytic geometry; an elementary textbook. -X V easily seen (see Fig. 8) ^ to be X = X0 + X1, 2/ = 2/o 4- y9 [12] z = So + z . 10. Transformation of coordinates from one set of rec-tangular axes to another which has the same origin. — Letiav Pv 7i)> (a2> && 72) an(^ (a3 $3* 73) ^e the directionangles of OX, OY\ andOZ* with respect to theoriginal axes. The coor-dinates (x, y, z) of anypoint P are the projec-tions of OP on OX, OY,and OZ. But the brokenline made up of xf, y\ andz extends from 0 to P,and will therefore havethe same projections on *the axes as OP. Hence(by A
. Plane and solid analytic geometry; an elementary textbook. -X V easily seen (see Fig. 8) ^ to be X = X0 + X1, 2/ = 2/o 4- y9 [12] z = So + z . 10. Transformation of coordinates from one set of rec-tangular axes to another which has the same origin. — Letiav Pv 7i)> (a2> && 72) an(^ (a3 $3* 73) ^e the directionangles of OX, OY\ andOZ* with respect to theoriginal axes. The coor-dinates (x, y, z) of anypoint P are the projec-tions of OP on OX, OY,and OZ. But the brokenline made up of xf, y\ andz extends from 0 to P,and will therefore havethe same projections on *the axes as OP. Hence(by Art. 5). Fig. 9. X = X COS ai + y COS a2 + Z COS a3,y = x cos pi 4- y cos (32 + z cos p3,2 = x cos 71 + 2/ cos -y- 4- s cos 73. [13] Ch. I, § 10] COORDINATE SYSTEMS. THE POINT 207 Let the student show that the transformation of coordi-nates cannot alter the degree of an equation. (See Part I.) PROBLEMS 1. What will be the direction cosines of OX, OY, and OZreferred to the new axes in Art. 10 ? 2. What six relations hold between al9 ft, y]5 a2, ft, etc.,from [7] ? 3. What six relations hold between alf ft, y1} a2, ft, etc.,from [10] ? 4. Show that the twelve relations obtained in problems 2and 3 are equivalent to only six independent conditions. Howmany of the coefficients in equations [13] are independent ? CHAPTER IILOCI 11. Equation of a locus. — If a point moves in spaceaccording to some law, it will generate some locus. As,for example, a point keeping at a fixed distance from afixed point will generate the surface of a sphere. If wecan translate the statement of the law into an a
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