GEOMETRY. 



Then fince the planf, whofe equation \s required, paffes Hence it follows that the perpendicular drawn from th^ 

 through the given point, and likewife through the point where origin of the co-ordmates uj on a plane whofe equation u 



the given line interfcfts the plane xy, the co-ordinates to 

 which points are 2 = o, * = k, > = i=. If the equation 

 to the plane be fuppofed 



% = Ax + By + 1), 

 in- which A, B, D, are co-efficients to be determined, then 



a'= Aa'+ By -I-D (i) 



o = Aa + B£:-f-D (2) 



Now the right lines being in the plane, fuppofe them both 



-f i_y + E =i Cisexprefiedby — ;= 



-¥i 



I + a- 

 Having the equations of a (Iraight line -j "* 



= a z -f a 1 



the equation of a plane perpendicular to this line drawn 

 through the point x', y', a, is a {x — x ) + b [y — y') 

 -f z — a' ;= o. 



To find the co-ordinates of the point of interfedlion of the 



moved parallel to themfelves till the plane paffes through the . ^^^ equations of the ftraight'line may be put under the 

 origin of the co-ordinates, the equations there will be, tor r . . .-1 



,ay 



= bi 



■'g 

 the llraight hne, 



X ^ a X, y =■ b %: 



and for the plane .r = A .v -)- B ^ 



In this pofition the line is Hill in the plane, fo that their 



co-ordinates are ftill the fame ; therefore, 



z = Aflz -I- B3z,andi = Aa -t- Bi (3) 



The equations I, 2, 3, will give A, B, D in terms of 



a, b, a, /S, and the equation of the plane will be 



(x — x') (f — bz' -0) - (y—y')(x'-az-o,) 

 + (z-2') (^(bx'-u)-aW-^)^=o. 



Problem IV. 



Given the equations of a ftraight line, and of a plane to 

 determine the conditions ; I ft, that the plane and ftraight 

 line may be reftangular ; 2d. The co-ordinates of the points 

 where they meet ; 3d. The diftance of this point from a given 

 point, either in the given line or given plane. 



When a plane is perpendicular to a ftraight line, the inter- 

 feftion of the plane with the co-ordinate planes and the pro- 

 jeftion of the line with thefe fame planes are perpendicular 

 to each other. 



Letx— az + a,y= la -(- ^, be the equations to the line ; 

 a = A .r + Bji -f C the equation to the plane ; the equations 

 to the interfeaions of this plane with the reftangular planes 



of.f2,and.fjr, arez — A.r-|-C z = B_y + C, but the 



plane being perpendicular to the line isA = — a>B= ~*' 

 therefore the equation of a plane perpendicular to the line, 

 ii ax + by + z=C: combining this equation with thofe of 

 the ftraight line x = a z + a, y — b z + ^ v.'e niay deduce 

 the values of x,y, z, the co-ordinates of the point in which 

 the ftraight line intcrfedts the plane. If the plane is given by 

 the equation ax + by + z = C, and the perpendicular to it 

 be required to be drawn tiirough a point whofe co-ordinates 

 are X, y, z, the equation to this perpendicular will be 



X — x' = aiz - z') y — y' z= b {z — z') a.nd the 



equation to the plane may be expreflied in this form 

 M (* -x) + b{y -y ) + z-z' = C-nx'- by' - z' 



Let X, Y, Z, be the co-ordinates of the point of interfec- 

 tjon of the plame and perpendicular, then 



Z = ,-+C-...'-*,-z' 



Y=y + 



X = 



i+a' + b- 

 b{C — ax — by' -zf 

 I + a= -f *' 

 a{C - ax' — by' — z> 



1 + a' + b- 



The length of the perpendicular comprehended between 

 t}ke points X, Y, ^ and the points x',y', z 'is = 

 Vi^-x-r+{\-fy + {Z-z'r 



^r 



_ C-ax'-by'-z' 



V I 



+ «° + i' 



Z = 



Y = -^" 



X = 



following form ; 



X — x' = az -i- a — x' 



y -y' =bz + e -y' 



Let X', Y', Z', be the co-ordinates of the points of in- 

 terfedlion ; then 



c W - «) + ^ (> ' - <9) + g ' 



1 + a' + i- 

 I { a (x' -u) .^biy'-IS) + z' 



I + a' + b' 

 a [a {x -c) + i (y - /3) + -' 



1 -t- a + A- 

 Subftituting for X', Y', Z', their values in the radical 



x/Tx' - *•)- + (Y -yy -r (Z - z'y 



an expreffion is obtained for the perpendicular contained be- 

 tween the given point of the ftraight line, of which the co- 

 ordinates are X ,Y',Z'. When the ftraight line paffes through 

 the origin of the co-ordinates, its equations become x — a z, 

 y =^ b z 



'X'^ + Y'^ TtJ- 



the ftraight line, drawn from the origin of the co-ordinates 

 to its interfeftion with the perpendicular let fall from 

 on this fuppofuion, a = o, /3 =r o 



and the radical ,t—-; — 



expreffes tie length of 



the point x',y', z', upon it : 

 „, a x' -{ by' -^ % 



1 + a^ 



+ b- 



Y' = bz' 



fore ^/ X" 



Y'= + Z" = ' 



. X' = ax: 

 ax -\- b y' -Y z 



there- 



\/ 1 + fl- + ^' * 

 This expreflion is ufed in finding the angle which two 

 ftraight lines make with each other. 



4. The equation of two ftraight lines being given to find 

 the angle which they make with each other, and if they do 

 not interfeft each other to determine the angle which their 

 projeftions form on a plane that is parallel to them, let 

 the equations of the given lines be 



r.ix-az f « 



^'Mj- =iz + /S 



1 f * = a' z + a.' 



'''^iy^bz + fi' 



If they interfeft, the angle which they make is equal to 

 the angle formed by their parallels which pafs through 

 the origin of the co-ordinates; the equations to their parallels 



being < , z- ^ • If a point be taken on the 



fecond parallel, whofe co-ordinates are x', v', z', and a per- 

 pendicular be let fall from this point to the firft parallel ; 

 then in the right-angled triangle formed by this perpendicu- 

 lar, and by the ftraight lines drawn from the origin of the 

 co-ordinates to the two extremities of this perpendicu- 

 lar, there are given the two fides which contain the re- 

 quired angle ; the expreffion for one of thefe fides is 



\~x^T 



