Consider the planes given by the equations

Question

Consider the planes given by the equations

−(x+2y+4z)=1,

4x+2y+z=7.

(a) Find a vector v^-> parallel to the line of intersection of the planes.

v^-> =

(b) Find the equation of a plane through the origin which is perpendicular to the line of intersection of these two planes.

This plane is

Answer 1

A vector parallel to the intersection of the two planes is the cross product of their normal vectors.

Let u = <1, 2, 4 > the normal vector of the first plane and v = < 4, 2 ,1> the normal vector of the second plane.

Then their cross product w =< -6, 17, -6 >.

The vector w or any scalar multiple of it is parallel to the intersection.

B. Find the equation of a plane through the origin which is perpendicular to the line of intersection of these two planes.

This plane is has as normal the vector w and passes through the origin

Then its equation is - 6(x -0 ) +17 (y -0 ) -6( z-0) =0

6x - 17y +6z = 0