GEOMETRY 55 



2.625 The perpendicular distance from the point # 2 , ^2, z 2 to the line: 



x - xi _ y - y\ _ z zi 



li mi m 



is: 



d = { (x 2 - *i) 2 + (y* - yiY + (z 2 - zi) 2 }* - {Ii(x 2 - xi) + mi(y 2 - yi) + i(s. - Zi)} . 



2.626 The direction cosines of the line passing through the two points Xi, yi, z\ 



and #2, yz t z 2 are: 



fa - xi), (j 2 - yi), (zz - zi) 



2.627 The two lines: 



x = miz + pi, x = nhz + p 2 , 



and 

 y = mz + qi, y = thz + q 2j 



intersect at a point if, 



(mi - mz) (qi - q z ) - (HI - th) (pi - p 2 ) = o. 

 The coordinates of the point of intersection are: 



p 2 - pi 



mi mz ' HI nz ' m\ m^ n\ 



The equation of the plane containing the two lines is then 



(HI - th) (x - miz - pi) = (mi - w 2 ) (y- n\z - qi). 



SURFACES 



2.640 A single equation in x, y, z represents a surface: 



F(x, y, z) = o. 



2.641 The direction cosines of the normal to the surface are: 



dF_ dF dF 



~dx' dy dz 



I, m, n = 



2.642 The perpendicular from the origin upon the tangent plane at x, y, z is: 



p = Ix + my + nz. 



2.643 The two principal radii of curvature of the surface F (x, y, z) = o are 

 given by the two roots of: 



