Plane and solid analytic geometry; an elementary textbook . he line makes with the axis. The projection, AB, of a directed line, AB, is evidentlya directed line. If a broken line AB, BC, CD is pro-jected on an axis, the algebraic sum of the projectionsof its parts will be the distance along the axis from theprojection of A to the projection of D. The projection onany axis, then, of any closed path in space, whichis looked upon as generated by the movement of a pointfrom A to B, B to C, etc., is zero. 6. Polar coordinates. — Let OX, OY, OZ be a set ofrectangular axes in space, and let P
Plane and solid analytic geometry; an elementary textbook . he line makes with the axis. The projection, AB, of a directed line, AB, is evidentlya directed line. If a broken line AB, BC, CD is pro-jected on an axis, the algebraic sum of the projectionsof its parts will be the distance along the axis from theprojection of A to the projection of D. The projection onany axis, then, of any closed path in space, whichis looked upon as generated by the movement of a pointfrom A to B, B to C, etc., is zero. 6. Polar coordinates. — Let OX, OY, OZ be a set ofrectangular axes in space, and let P be any point. DrawOP. The position of P is evidently determined if we knowits distance p from the origin, and the angles a, /3, and 7which OP makes with the coordinate axes. The distance p is called the radius vector of the point Ch. I, § 6] COORDINATE SYSTEMS. THE POINT 213 P; «, /3, and 7, the direction angles of the line OP; andthe four quantities (jo, «, /3, 7), the polar coordinates of thepoint. Cos a, cos /3, and cos 7 are called the direction. Fig. 5 cosines of the line OP, and may be represented by theletters Z, ???., and n. Let L, M, and JV be the projections of P on the OL = x, 031=7/ and ON=z; and from right tri-angles we have X = p COS a, y = p COS P, [6] » = p COS Y. These equations give the relations between the rec-tangular and polar coordinates of any point. Squaringand adding, we have x2 + y2 -f 22 = p2 (cos2 a + cos2 ft + cos2 7). But from [2], p2 = x2 + y2 + z2. Hence cos2 a + cos2 p + cos2 y = !• [7] 214 ANALYTIC GEOMETRY OF SPACE [Ch. I, § 6 That is, the sum of the squares of the direction cosinesof any line is unity. Hence the four quantities used aspolar coordinates of a point are equivalent to only threeindependent conditions, as we should expect. We may always choose these coordinates so that theyshall all be positive and so that the angles «, /3, y shallnot be greater than 180°. since any line parallel to OP makes the same anglesw
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