Find the normal equation of the plane that contains the line

Find the normal equation of the plane that contains the line [x1, x2, x3] = [4, 0, 1] + s[3, 1, 5] and is orthogonal to the plane x1 + x2 + x3 = 7.

Solution

Equation of plane orthogonal to x1 +x2 +x3 =7

Direction vector of line u = < 3, -1 , 5>

v = < 1, 1 , 1 > the normal vector of the plane

Vector product of u x v = < -6, 2 4>   This is the normal to the plane we are looking for.

Equation of plane : -6( x1 -4) +2 ( x2 -0) + 4(x3 -1) =0

-3x1 +12 +x2 +2x3 -2 =0

3x1 - x2 -2x3 = 10

Normal form : divide by sqrt( 3^2 +1 +2^2) = sqrt( 9+1+4) = sqrt15

Normal equation of plane : 3x1/sqrt15 -x2/sqrt15 -2x3/sqrt15 = 10/sqrt15