A plane passes through the three points 1 2 1 4 3 1 and 7 4

A plane passes through the three points (1, 2, 1), (4, 3, 1) and (7, 4, 2) (a) Find the parametric equation of the plane. (b) Find the normal equation of the plane.

Solution

Let P = ( 1,2 ,-1) and Q = ( 4,3,1) and R = ( 7,4,2)

So, we have vector PQ = ( 3, 1, 2)

PR = ( 6 , 2, 3)

Cross product of PQ x PR = ( -1,3 , 0)

Do, PQ x PR perpedicular to PX

< -1,3,0>.( x-1, y-2, z+1> =0

-1(x-1) + 3(y-2)+ 0(z+2) =0

-x +1 +3y -6 =0

a)   3y -x -5 =0

b)   Normal form : divide by sqrt( 3^2 +1+5^2) = sqrt( 9 +1+25) = sqrt26

(3y -x -5)/sqrt(26) =0