. Differential and integral calculus, an introductory course for colleges and engineering schools. 21) and (x2, y2, z2) (Art. 181). Hence thisline lies wholly in the surface (a). Consequently the surface (a)is such that the line joining any two points of it lies wholly in thesurface, and such a surface is a plane. Q. E. D. The line of intersection of a surface with a coordinate planeis termed the trace of the surface in that coordinate plane, andthe equation of the trace is obtained by putting x = 0, or y = 0,or z = 0, as the case may be, in the equation of the surface. Forexample, the trace o
. Differential and integral calculus, an introductory course for colleges and engineering schools. 21) and (x2, y2, z2) (Art. 181). Hence thisline lies wholly in the surface (a). Consequently the surface (a)is such that the line joining any two points of it lies wholly in thesurface, and such a surface is a plane. Q. E. D. The line of intersection of a surface with a coordinate planeis termed the trace of the surface in that coordinate plane, andthe equation of the trace is obtained by putting x = 0, or y = 0,or z = 0, as the case may be, in the equation of the surface. Forexample, the trace of the surface 2x2 -Zxy -y2 - 4z2 = 5 in the ^-plane is obtained by putting y = 0 in the equation ofthe surface, and is 2x2 -±z2 = traces of the plane(a) ax + by + cz + d = 0are right lines, and their equa-tions in the x?/-plane, the yz-plane, and the zx-plane arerespectively Y zX ax + by + d = 0, by + cz. + d = 0, ax + cz + d = plane is indicated in a figure by drawing its 270 GEOMETRY OF THREE DIMENSIONS §184 If in the equation of the plane (a) we set y = 0, z — 0, we get ax + d = 0, whence x = , and this is the intercept of (a) on a the x-axis. Similarly, — ■=■ and — - are the intercepts of (a) on the u c y-Sixis and the 2-axis. Equation (a) may be thrown into the form _d^ d^ d l a b c and on placing —the form (i) d d , dffli 6i A = ci, equation (a) takes which is the intercept form of the equation of a plane. A right line in space cannot be given by a single equation. Butthe equation of any two planes that contain the line are regarded asthe equations of the line. 184. Exercises. 1. Find the mid-point and also the points of trisection of the linewhich joins the points ( — 2, 1, 3) and (3, —5, —4). 2. Find the points where the line 3 x + 2 y - 4 z + 7 = 0x-y + 3z-l = 0 pierces the coordinate planes. Draw a figure showing these points an ithe traces of the planes. 3. Find the vertices and the lengths of the edges of the tetrahedronwh
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