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Parallel Planes

Parallel Planes

Two distinct planes that do not intersect.

 

Two horizontal rectangular planes stacked parallel to each other with a gap between them, never intersecting.

Worked Example

Problem: Determine whether the planes 2x + 3y − z = 5 and 4x + 6y − 2z = 18 are parallel.
Step 1: Identify the normal vector of each plane. A plane written as ax + by + cz = d has normal vector (a, b, c).
Plane 1: n1⃗=(2, 3, −1)Plane 2: n2⃗=(4, 6, −2)\text{Plane 1: } \vec{n_1} = (2,\, 3,\, -1) \qquad \text{Plane 2: } \vec{n_2} = (4,\, 6,\, -2)
Step 2: Check whether one normal vector is a scalar multiple of the other. If so, the planes are parallel.
n2⃗=2⋅n1⃗=2(2, 3, −1)=(4, 6, −2)\vec{n_2} = 2 \cdot \vec{n_1} = 2(2,\, 3,\, -1) = (4,\, 6,\, -2)
Step 3: Since the normal vectors are scalar multiples, the planes face the same direction. Now check if they are distinct: divide Plane 2's equation by 2 to get 2x + 3y − z = 9, which differs from Plane 1's constant of 5.
5≠95 \neq 9
Answer: The planes are parallel (and distinct) because their normal vectors are proportional but the equations are not identical.

Why It Matters

Parallel planes appear frequently in coordinate geometry and physics. For example, the floor and ceiling of a room model two parallel planes. Recognizing parallel planes also helps when solving systems of three linear equations — if two of the planes are parallel, the system has no single point of intersection and therefore no unique solution.

Common Mistakes

Mistake: Concluding that two planes are the same plane when their normal vectors are proportional.
Correction: Proportional normal vectors only mean the planes face the same direction. You must also check the constant terms. If, after scaling, the equations are identical, the planes coincide (same plane) rather than being parallel and distinct.

Related Terms

  • Plane — The geometric object that parallel planes describe
  • Distinct — Parallel planes must be distinct, not coinciding
  • Parallel Lines — The analogous concept in two dimensions
  • Normal Vector — Used to test whether two planes are parallel
  • Skew Lines — Non-intersecting lines that are not parallel in 3D