If A B and C lie in the same plane, then any point P in the plane with coordinates #(x,y,z)# can be represented by:
#vec(AP) = mvec(AB)+nvec(AC) #
From the points given:
#vec(AP)=((x-1),(y-1),(z+1))color(white)(88)#, #vec(AB)=m((5),(3),(-4))color(white)(88)#,#vec(AC)=n((-6),(-3),(9))#
#:.#
#x=1+5m-6ncolor(white)(88)#[ 1 ]
#y=1+3m-3ncolor(white)(88)#[ 2 ]
#z=-1-3m+9ncolor(white)(88)#[ 3 ]
Eliminating #m and n#
Multiply [1] by #3/5# and add to [3]
#3/5x+z=-2/5+27/5n color(white)(88), 3x+5z=-2+27ncolor(white)(8)#[4]
Add [2] and [3]
#y+z=6ncolor(white)(88)# [5]
Multiply [ 5 ] by #-27/6# and add to [ 4 ]
#-27/6y+3x- 27/6z+5z=-2#
Cartesian equation of the plane
#color(blue)(18x-27y+3z=-12)#
The vector equation of the plane will be:
#Pi=((1),(1),(-1))+m((5),(3),(-4))+n((-6),(-3),(9))#
PLANE:
