By analyzing the normals, determine if the two planes intersect in a line, are parallel and distinct, or are coincident.
x+2y+3z-3=0
2x-y+z-7=0
To determine if the two planes intersect in a line, are parallel and distinct, or coincident, we can analyze their normal vectors. The normal vector of a plane is a vector that is perpendicular to the plane.
First, we need to find the normal vectors for each plane. The normal vector of a plane in the form of Ax + By + Cz + D = 0 is given by the coefficients A, B, and C.
For the first plane:
x + 2y + 3z - 3 = 0
The coefficients are A = 1, B = 2, and C = 3. Therefore, the normal vector of the first plane is (1, 2, 3).
For the second plane:
2x - y + z - 7 = 0
The coefficients are A = 2, B = -1, and C = 1. Therefore, the normal vector of the second plane is (2, -1, 1).
Now, let's analyze the normal vectors:
If the normal vectors are parallel, it means the planes are either parallel or coincident. If they are not parallel, it means the planes intersect.
To determine if the normal vectors are parallel, we can compute the cross product of the two normal vectors. If the cross product is the zero vector, it means the normal vectors are parallel.
Let's compute the cross product of the two normal vectors:
(1, 2, 3) × (2, -1, 1) = (6, 7, -5)
Since the cross product is not the zero vector, it means the normal vectors (and therefore the planes) are not parallel.
Therefore, the two planes intersect in a line.