What is the intersection of two distinct planes?
The correct answer and explanation is:
The intersection of two distinct planes is a line.
Explanation:
In three-dimensional space, planes are flat, two-dimensional surfaces that extend infinitely in all directions. When two planes are distinct (meaning they are not coincident or identical), they either do not intersect at all or intersect in a line. The case where they intersect in a line occurs when the planes are not parallel. If the planes are parallel, they will never meet and therefore have no intersection.
For two planes to intersect in a line, their equations must not be parallel. In mathematical terms, the normal vectors of the two planes must not be parallel. The normal vector of a plane is a vector that is perpendicular to the surface of the plane. If the normal vectors of the two planes are parallel, the planes are parallel and do not intersect. However, if the normal vectors are not parallel, the planes will intersect in a line.
To understand this geometrically, imagine two flat surfaces in space. If you tilt one of the surfaces, it will eventually cross the other surface at some angle. This crossing forms a straight line where the two planes meet. This line is where every point on it satisfies the equations of both planes simultaneously. Therefore, the intersection of two distinct non-parallel planes is always a line.
In summary, the intersection of two distinct planes in three-dimensional space is a line, assuming the planes are not parallel. If the planes are parallel, there is no intersection. The line of intersection can be found by solving the system of equations that describe the planes.