Plane and solid analytic geometry; an elementary textbook . to all possible cases. It might seemat first that in Fig. 12 the equation KPt=M1P1 — MXKdoes not hold. But if MXK is replaced by its equal— KMV the equation is at once seen to be true. Let the student draw various figures with the pointsin different quadrants, and assure himself that the samedemonstration holds for all. Care must be taken to readthe lines always in the proper direction. For simplicitythe figures will usually be constructed in the first quad-rant, but the student should always satisfy himself thatthere is no restrictio
Plane and solid analytic geometry; an elementary textbook . to all possible cases. It might seemat first that in Fig. 12 the equation KPt=M1P1 — MXKdoes not hold. But if MXK is replaced by its equal— KMV the equation is at once seen to be true. Let the student draw various figures with the pointsin different quadrants, and assure himself that the samedemonstration holds for all. Care must be taken to readthe lines always in the proper direction. For simplicitythe figures will usually be constructed in the first quad-rant, but the student should always satisfy himself thatthere is no restriction on their position, and that, if anyother figure is constructed and lettered in a correspond-ing way, just the same demonstration will hold letterfor letter. Oh. II, § 12] THE POINT 13 12. Distance between two points in oblique coordinates. —When the axes are oblique, draw M1P1 and M2P2, theordinates of Px and jP2, and the line P2K parallel tothe X-axis. Since P2K and KPX are to be expressedin terms of the coordinates of Px and P2, their positive. Fig. 13. Fig. 14. directions will be the same as the positive directions ofthe axes. The angle between them will always be a>,and the generalized form of the law of the cosines(Art. 7) gives PXP2 = Vp2jf2 + jzpf + 2 P2K- KPX cos *», where not only the magnitudes, but also the directionsof the lines are considered. But P2K= M2MX = 0M1 - 0M2 = xx- x2, and KP^M^-M^M^-MzP^y^yv Substituting these values, we haveP\P% = V(a?! - a?2)2 + (2/i - Z/2)2 + 2(«i - a?2)(2/i - y-i) cos «, [2]as the distance between two points in oblique coordinates. 14 ANALYTIC GEOMETRY [Ch. II, § 13 PROBLEMS 1. Find the distance between the two points whose rec-tangular coordinates are (— 2, 6) and (1, 5). Solution. —In using formulas [1] and [2] we may choose either ofthe points for Pi and the other for P2. Let (-2, 6) he the coordinatesof Pi, and (1, 5) the coordinates of P2. Then PiP2 = V(-2 - 1)*+(tt - 5)* = VlO. 2. Find the lengths of the sid
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