Question

for the system of three planes : 2x_2y+2z=4 4x_5y+8z=0 x_y+z=2 a) are any of the plane parallel justify your answer? b)determine if the normal are coplanar? what does this tell you about the system? c: solve the system if possible.

Answer 1

a) None of the planes are parallel. b) The normal vectors are not coplanar. c) The system of three planes does not have a unique solution.

Step-by-step explanation:

a) To determine if any of the planes are parallel, we can examine their normal vectors. Two planes are parallel if their normal vectors are parallel. Let's find the normal vectors of the three planes:

Plane 1: 2x - 2y + 2z = 4. The normal vector is [2, -2, 2].

Plane 2: 4x - 5y + 8z = 0. The normal vector is [4, -5, 8].

Plane 3: x - y + z = 2. The normal vector is [1, -1, 1].

Since none of the normal vectors are parallel, none of the planes are parallel.

b) To determine if the normal vectors are coplanar, we can check if their cross product is the zero vector. Let's calculate:

[2, -2, 2] x [4, -5, 8] = [-6, -4, -14].

The cross product is not the zero vector, which means the normal vectors are not coplanar.

c) Since the planes are not parallel and the normal vectors are not coplanar, the system of three planes does not have a unique solution.