. Plane and solid analytic geometry . If P lies on L, it is its own jirojectiouon L. Let PQ be any directed line-segment, and let L be an arbi-trary line. Let 31 and N be respectively the projections ofP and Q on L. The projectionof the directed line-segment PQon L shall be defined as the di-rected line-segment MN, or thenumber which represents MN al-gebraically. Since MN= — NM,it follows that Proj. PQ = -Pioj. QP. If PQ lies on a line perpendicular to L, the points M andN coincide, and we say that the projection MN oi PQ on L iszero. Such a directed line-segment 3fX, whose end-points areident
. Plane and solid analytic geometry . If P lies on L, it is its own jirojectiouon L. Let PQ be any directed line-segment, and let L be an arbi-trary line. Let 31 and N be respectively the projections ofP and Q on L. The projectionof the directed line-segment PQon L shall be defined as the di-rected line-segment MN, or thenumber which represents MN al-gebraically. Since MN= — NM,it follows that Proj. PQ = -Pioj. QP. If PQ lies on a line perpendicular to L, the points M andN coincide, and we say that the projection MN oi PQ on L iszero. Such a directed line-segment 3fX, whose end-points areidentical, we may call a nil-segment; to it corresponds thenumber zero. It is evident that in taking the sum of a num-ber of directed line-segments, any of them which are nil-segments may be disregarded, just as, in taking the sum of a set of numbers, any of themwhich are zero may be disre-garded. Consider an arbitrarybroken line PP^P. ■ ■ • P„ its projection on L ismeant the sum of the pro-jections of the directed line-. ill M^ ilf J il/3 n-2 M^.iN Fig. 3segments PPi, P1P2, • • •, Pn-iQ, or 3f3f, + 3f,3f2 -f- • • • + 3f„_rN. This sum has the same value as 3fN, the projection on Lof the directed line-segment PQ; cf. § 1, (3): JOfi 4- 3Ii3L -\- ■•■ + M„_,y= 3fN. 6 ANALYTIC GEOMETRY Hence the theorem: Theorem 1. The sum of the projections on L of the segmentsPPi, PiP-i, ■ • -, Pn-\Q 0/ ^ broken line joining P with Q is equalto the projection on L of the directed line-segment PQ. If, secondly, the same points P and Q be joined by anotherbroken line, PPlPl, ••• PL-iQ, the projection of the latter onL will also be equal to MN: MM[ + M[M^2-\- ■ + Ml_^N= MN. Hence the theorem: Theorem 2. Given two broken lines having the same extremi- ties PP,P2 ■ P^.iQ and PP[PlPL_,Q. Let L be an arbitrary straight line. Then the sum of the pro-jections on L of the segments PP^, P1P2, • • •, Pn-^Q, of which thefirst broken line is made up, is equa
Size: 2233px × 1119px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1920, bookidplanesolidan, bookyear1921
