Find the vector equations of the plane passing through the points `R(2, 5, - 3)`, `S ( -2, -3, 5)`and `T(5, 3, -3)`

Explanation

Three given points are `R(2, 5, -3)`, `S(-2, -3, 5)`and `T(5, 3, -3)`
general equation of plane passing through the R(2,5,-3) is
` a(x-2)+b(y-5)+c(z+3) =0`....(i)
`:' S(-2,-3,5)` and `T(5,3,-3)`
passes through (i) then
`a(-2-3)+b(-3-5)+c(5+3) =0 => -4a -8b+8c =0 => a-2b+2c=0`....(ii)
`a(5-2)+b(3-5)+c(-3+3) =0 => 3a-2b+0c = 0` ...(iii)
by cross multiply (ii) and (iii) , we get
` a/(0+4) = b/(6-0) = c/(2+6) = k`
`=> a=2k , b=3k , c=4k
`:. = put in equation (i)
` 2k(x-2)+3k(y-5)+4k(z+3) =0`
` 2x-4+3y-15+4z+12=0`
`2x+3y+4z=7`
vector equation of the plane is `vecr.(2hati+3hatj+4hatk) = 7`