Find the equations of the planes that passes through three points. (a) `( b ) (c)(( d ) (e)1," "1," "" "1( f ))," "(( g ) (h)6," "4," "" "5( i ))," "(( j ) (k) " "4," "" "2," "3( l ))( m )` (n) (o) `( p ) (q)(( r ) (s)1," "1," "0( t ))," "((

Explanation

(a) Given points are `A(1,1,-1), B(6,4,-5)` and `C(-4,-2,3)`.
First we check whether the points are collinear.
`:. |{:(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3)):}|=|{:(1,1,-1),(6,4,-5),(-4,-2,3):}|`
`= 1 (12-10) - 1(18-20)`
`-1(-12+16)`
`= 2+2-4 = 0`
Since gi ven three points are collinear. Therefore, number of planes passing through three points are infinite.
(b) Given points are `A(1,1,0), B(1,2,1)` and `C(-2,2,1)` .
First we check whether the points are collinear.
`|{:(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3)):}|=|{:(1,1,0),(1,2,1),(-2,2,-1):}|`
` =1(-2-2)-1(-1+2)+0(2+4)`
` = - 5 ne 0`
Therefore , three points `A,B` and C are not collinear. Equation of plane passing thro ugh three points `(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2))` and `(x_(3),y_(3),z_(3))`
`|{:(x-x_(1),y-y_(1),z-z_(1)),(x_(2)-x_(1), y_(2)-y_(1),z_(2)-z_(1)),(x_(3)-x_(2),y_(3)-y_(2),z_(3)-z_(2)):}|=0`
`rArr |{:(x-1,y-1,z),(0,1,1),(-3,0,-2):}|=0`
`rArr -2(x-1)-3(y-1)+3z=0`
`rArr -2x+2-3y+3+3z=0`
`rArr 2x+3y-3z-5=0`
`rArr 2x+3y-3z=5`
which is the required equation of plane.