a) Find a vector b)Find the equation of a plane

Question

Consider the planes given by the equations

2x−3y−5z=2,

2x−2y+3z=4.

(a) Find a vector v→ parallel to the line of intersection of the planes.

(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

Solving the system of the two equations for z we get

x = 4 - (19/2)z

y = 2 - 8z

z =z

That is a vector parallel to the intersection of the two planes

is u =< -19/2 , -8, 1 > or any scalar multiple

say n = < -19, -16, 2 >

The plane passing through the origin ( 0, 0, 0 )

has equation -19x -16y +2z =0