Parallel Planes and Points (3D)

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SUMMARY

Point A (3,2,-1) is equidistant from two parallel planes, P1 and P2. The equation of plane P1 is given as 2x - y + 2z + 4 = 0. Using the distance formula D = |ax_o + by_o + cz_o + d| / √(a² + b² + c²), the distance from Point A to P1 is calculated to be 2. Consequently, the equation for plane P2 can be derived by adjusting the constant term in the equation of P1 by the calculated distance, resulting in P2's equation being 2x - y + 2z + 2 = 0.

PREREQUISITES
  • Understanding of 3D geometry and coordinate systems
  • Familiarity with the distance formula for planes
  • Knowledge of parallel planes and their properties
  • Basic algebra for manipulating plane equations
NEXT STEPS
  • Study the derivation of plane equations in 3D geometry
  • Learn about the properties of parallel planes and their equations
  • Explore applications of the distance formula in various geometric contexts
  • Practice solving problems involving points and planes in three-dimensional space
USEFUL FOR

Students studying geometry, educators teaching 3D coordinate systems, and anyone interested in understanding the relationships between points and planes in three-dimensional space.

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Homework Statement



Point A (3,2,-1) is equidistant from two planes (parallel), known as P1 and P2. P1 has the equation 2x-y+2z+4 = 0. Find the equation of P2.


Homework Equations



D = abs val(ax_o + by_o + cz_o + d) divided by sqrt (a^2 + b^2 + c^2)

The Attempt at a Solution



I used the distance formula and figured out that the distance between the point and either plane is 2. But what do I do now? I know that the plane I'm looking for, since it's parallel, has a perpendicular vector that is the same as P1, if not a scalar multiple. But what next? Thanks in advance.
 
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Compute a point on the other plane.
 
how do i do that?
 

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