Plane and solid analytic geometry; an elementary textbook . OBLEMS 1. Write the equations of each of the coordinate axes. 2. Write the most general form of the equations of a line ineach of the coordinate planes; parallel to each of the coordi-nate planes; parallel to each of the coordinate axes. 3. Show how to find the points where a given line piercesthe coordinate planes, and by this means plot the lines inproblem 4. Ch. IV, § 23] THE STRAIGHT LINE 237 4. Reduce these equations to their simplest forms (a) 2a?-3# + z - 6 = 0, (6) 2x + 3y x + 7/ — 3 z — 1 = 0.(c) 2a? + 4y + 3« + 6 = 0, 3a? +
Plane and solid analytic geometry; an elementary textbook . OBLEMS 1. Write the equations of each of the coordinate axes. 2. Write the most general form of the equations of a line ineach of the coordinate planes; parallel to each of the coordi-nate planes; parallel to each of the coordinate axes. 3. Show how to find the points where a given line piercesthe coordinate planes, and by this means plot the lines inproblem 4. Ch. IV, § 23] THE STRAIGHT LINE 237 4. Reduce these equations to their simplest forms (a) 2a?-3# + z - 6 = 0, (6) 2x + 3y x + 7/ — 3 z — 1 = 0.(c) 2a? + 4y + 3« + 6 = 0, 3a? + 6y + 22-l = 0. 3t/ (e) 2a>-3y- z + 2 = 0, (/) 4a?-6y + 32J-l = 0. 2t/ 6 z - 12 = 0, 4# — t/ + 12z + 4 = 0.(rt) 4y+ 3*+ 1=0, — 2 12 = 0. 3z- 2 = 0, 2 + 4 = 0. 5. Find the equations of the line of intersection of the plane2x — 3y -\- z — 6 = 0 with the coordinate planes. 23. The equations of a line in terms of its directioncosines and the coordinates of a point through which itpasses. — Let «, /3, and 7 be the direction angles of the. Fig. 13. c line and Pt a point through which it passes. Let P beany point on the line. Then from the figure x — xx = PXP cos a, y-yl = PXP cos ft z — z1 = PXP cos 7. 238 ANALYTIC GEOMETRY OF SPACE [Ch. IV, § 24 Solving these for PXP, and equating the values, we haveaj - asi y - y\ z - Z\ cos a cos 0 COS X PROBLEMS [22] 1. What form will these equations take when a = 90° ?when a = 90°, and £ = 90° ? 2. Find the equations of a line through the point ( — 1,2, -3) if (a) a = 60°, /? = 60°, y = 45°; (6) a = 120°, (3 = 60°, y = 135°j (c) cos a = | V3, cos /3 = i, cos y = 0. Show that the given values are possible in each case and plotthe line. 3. Find the equations of a line through the origin, equallyinclined to the axes. 24. Given the equations of a line, to find its directioncosines. — The method is best shown by an example. Letthe equations of a line, reduced to their simplest form, be x = 5 z — 6, and y =
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