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Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1909 Excerpt: ...= 0, formed by equating the first two fractions. 105. Equations of a tangent line to a space curve. is any given point of a space curve, and P% (xl + Aa;, y1 + Ay, zl + As) is any second point of the curve, the equations of the secant P are ZJ (§io4) or where As is the length of the arc P. Denning the tangent at P± as in I, § 59, we have as its equations 106. Equations of a straight line in terms of its direction cosines and a known point upon it. Let Pl(xv yv zj (fig. 65) be a known point of the line, and let /, m, and n be its direction cosines. Let P(x, y, z) be any point of the line. On IP as a diagonal construct a parallelopiped as in § 93. Then if we denote PlP by r, we have These are the parametric equations of the line, the variable param eter being r, r being positive if the point is in one direction from Pv and negative if it is in the other direction from Pi. By eliminating r we have (2) I m n 65 which are but two independent linear equations. 107. Problems on the plane and the straight line. In this article we shall solve some problems illustrating the use of the equations of the plane and the straight line. 1. Plane determined by three known points. Let the three given Q. (1) Since Pv P2, and P3 are points of the plane, their coordinates satisfy (1). Therefore Q, (2) Q, (3) Q. (4) We may now solve (2), (3), and (4) for the ratios of the unknown constants A, B, C, and D, and substitute in (1), or we may eliminate A, B, C, and D from the four equations (1), (2), (3), and (4). By either method the equation of the required plane is found to be Ex. 1. Find the equation of the plane determined by the three points (1,1, 1), (-1,1, 2), and (2,-3,-1). The required equation is If the three given points are on the same straight line, n...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1909 Excerpt: ...= 0, formed by equating the first two fractions. 105. Equations of a tangent line to a space curve. is any given point of a space curve, and P% (xl + Aa;, y1 + Ay, zl + As) is any second point of the curve, the equations of the secant P are ZJ (§io4) or where As is the length of the arc P. Denning the tangent at P± as in I, § 59, we have as its equations 106. Equations of a straight line in terms of its direction cosines and a known point upon it. Let Pl(xv yv zj (fig. 65) be a known point of the line, and let /, m, and n be its direction cosines. Let P(x, y, z) be any point of the line. On IP as a diagonal construct a parallelopiped as in § 93. Then if we denote PlP by r, we have These are the parametric equations of the line, the variable param eter being r, r being positive if the point is in one direction from Pv and negative if it is in the other direction from Pi. By eliminating r we have (2) I m n 65 which are but two independent linear equations. 107. Problems on the plane and the straight line. In this article we shall solve some problems illustrating the use of the equations of the plane and the straight line. 1. Plane determined by three known points. Let the three given Q. (1) Since Pv P2, and P3 are points of the plane, their coordinates satisfy (1). Therefore Q, (2) Q, (3) Q. (4) We may now solve (2), (3), and (4) for the ratios of the unknown constants A, B, C, and D, and substitute in (1), or we may eliminate A, B, C, and D from the four equations (1), (2), (3), and (4). By either method the equation of the required plane is found to be Ex. 1. Find the equation of the plane determined by the three points (1,1, 1), (-1,1, 2), and (2,-3,-1). The required equation is If the three given points are on the same straight line, n...
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