Parallel planes
Parallel planes are two or more flat surfaces that do not intersect, no matter how far they are extended
Parallel planes are two or more flat surfaces that do not intersect, no matter how far they are extended. In other words, they are always the same distance apart and never touch.
To determine if two planes are parallel, we use the concept of slope. The slope of a plane is determined by its normal vector. A normal vector is a vector that is perpendicular to every vector lying in the plane.
Suppose we have two planes with equations:
Plane 1: Ax + By + Cz + D1 = 0
Plane 2: Ax + By + Cz + D2 = 0
If the normal vectors of both planes are parallel, then the planes are also parallel. Two vectors are parallel if they are scalar multiples of each other.
To find the normal vectors of the planes, we look at the coefficients A, B, and C. The normal vector of Plane 1 is (A, B, C) and the normal vector of Plane 2 is also (A, B, C).
Now let’s consider an example to determine if two planes are parallel:
Plane 1: 2x + 3y + 4z – 5 = 0
Plane 2: 4x + 6y + 8z + 10 = 0
The normal vector of Plane 1 is (2, 3, 4) and the normal vector of Plane 2 is (4, 6, 8).
To check if they are parallel, we can observe that (4, 6, 8) is twice the normal vector (2, 3, 4), so the planes are parallel.
Remember, if the normal vectors of two planes are scalar multiples of each other, then the planes are parallel.
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