. Plane and solid analytic geometry; an elementary textbook. are parallel, they must make the same angleswith the axes, and the conditions for parallelism are - ai =aj, p! = po, and yi = Y2« [11] PROBLEMS 1. Show that the three lines whose direction cosines are 12 -3 -4. _4_ 12 _3_. nrif1 3 -4 1213 13 13 13 13 13 dlUA 13 1 3 TT are mutually perpendicular. 2. Show that (3, 30°, 60°, 90°), and (5, 30°, 90°, 60°) arepossible polar coordinates of two points, and find the angle theysubtend at the origin. 3. Show that the conditions for parallelism are consistentwith [9] when 0 = 0°. 4. Find the rec
. Plane and solid analytic geometry; an elementary textbook. are parallel, they must make the same angleswith the axes, and the conditions for parallelism are - ai =aj, p! = po, and yi = Y2« [11] PROBLEMS 1. Show that the three lines whose direction cosines are 12 -3 -4. _4_ 12 _3_. nrif1 3 -4 1213 13 13 13 13 13 dlUA 13 1 3 TT are mutually perpendicular. 2. Show that (3, 30°, 60°, 90°), and (5, 30°, 90°, 60°) arepossible polar coordinates of two points, and find the angle theysubtend at the origin. 3. Show that the conditions for parallelism are consistentwith [9] when 0 = 0°. 4. Find the rectangular coordinates of the points inproblem 2. 5. Find the polar coordinates of the point (3, — 6, 2). 6. Find the angle subtended at the point (1, 2, 3) by thepoints (2, 3, 4) and (o, 4, 3). 206 ANALYTIC GEOMETRY OF SPACE [Ch. I, § 10 9. Transformation of coordinates. Parallel axes. — If the new axes are parallel to the old, and the coordinates of the new origin, re-ferred to the old axes, are(#0, y0, z0), the equationsof transformation are. -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)
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