Two planes that do not intersect are _____parallel

Two planes that do not intersect are called parallel planes.

To determine if two planes are parallel, we need to compare their normal vectors. The normal vector of a plane is a vector that is perpendicular (orthogonal) to the plane. If two planes have the same normal vector or their normal vectors are scalar multiples of each other, then the planes are parallel.

To find the normal vector of a plane, we can use the coefficients of the plane's equation. The general equation of a plane in three-dimensional space is Ax + By + Cz + D = 0, where A, B, and C are the coefficients, and (x, y, z) are the coordinates of a point on the plane.

For example, consider two planes with the following equations:
Plane 1: 2x - 3y + 4z - 5 = 0
Plane 2: 4x - 6y + 8z + 7 = 0

To find the normal vectors, we take the coefficients of x, y, and z in each equation. For Plane 1, the normal vector is (2, -3, 4), and for Plane 2, the normal vector is (4, -6, 8).

Next, we compare the normal vectors. In this case, the normal vector of Plane 1 is a scalar multiple of the normal vector of Plane 2. Specifically, we can multiply Plane 1's normal vector by 2 to get the same result as Plane 2's normal vector.

Therefore, we can conclude that the two planes are parallel.