By Smith, Charles, 1844-1916
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Extra resources for An elementary treatise on solid geometry
Let be the middle point of CO the shortest distance draw between the two straight lines CD, Through OA, OB parallel to CD, C D and let OX, OY bisect the angle A OB. Take OX, OY,OG for axes of co-ordinates x tana then, if AOB be 2a, the equations of OA, OB are y 2 = 0, and y = x tan a, z 0. , CD . t ; y Hence the equations of the parallel x tan a, z= c; and y = x tan a, s = When it rectangular, tions of CD, W CD Also . lines G are not of importance that the axes should be for axes: the equa jy will then be y = 0, ^ = c and # = 0, z = c.
Cosines are therefore | \/2, |\/2, ^ Ex. 2. Find in a The 3. Am. %(x- Find the direction-cosines of the . Ex. (2, 3, 4) 4. Hence the symmetrical form the equation of the line 3x+y~7z=Q. Ex. direction- actual values of the direction- line 5) x-2y- 5, =y=z- If. whose equations are Ans. -^. Write down the equation of the straight line through the point whicli is equally inclined to the axes. Ans. x-2 = y-3 z-4. THE STRAIGHT 16 LINE. To find the equations of a straight line through two 23. given points.
Acute angle or in the obtuse ? x- x-2y-z= cut ? Is the Is the point (l,-3, 1) in the 1 Ans. cos~ |-v/2, acute, obtuse. Ex. 3. Find the equation of the plane through (1, 4, 3) perpendicular to the line of intersection of the planes 3x + 4y + 7z + 4 0, and + 2z+3 = also of the plane through (3, 1, - 1) perpendicular to the line of intersection of the planes 3x + y-z = 0, 5x-3y + 2z=Q. = Ans. 15x + y-7z + 2=Q. Ex. 4. Shew that the line +iiz+p = Q if IX ^= \ y - -- fj. + m/jL + nv =0, v is parallel to x-y Ans.