Parallel Planes: Same/Opposite Sides of Origin?

In summary, the conversation discusses how the equation of a plane can determine its position relative to the origin. If two planes have the same sign for the value of d, they are on the same side of the origin, while if they have different signs, they are on opposite sides. This applies not only to parallel planes, but also to non-parallel planes, where the definition of "same side of the origin" may vary. Additionally, it is noted that multiplying both sides of the plane equation by -1 will not change its position, as long as the constants a, b, and c remain constant or proportional and do not change signs.
  • #1
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According to my book, if we write the equation of a plane as:

##ax + by + cz = d##

And two planes have values of ##d## with the same sign, they are on the same side of the origin. If they have values of ##d## with different signs, they are on opposite sides of the origin.

I'm confused as to whether this applies strictly to two parallel planes, or any pair of planes.
 
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  • #2
You must decide what "same side of the origin" means for non-parallel planes.
 
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  • #3
jbriggs444 said:
You must decide what "same side of the origin" means for non-parallel planes.

What about parallel planes? If both sides of the plane equation (cartesian form) are multiplied by -1, the position of the plane is unaltered, right?
 
  • #4
Right. In order to make the "parallel plane" version work, you need for a, b and c to either remain constant or at least to remain proportional and not change signs.
 
  • #5


The statement in your book applies to any pair of planes, not just parallel planes. The value of d in the equation represents the distance of the plane from the origin in the direction of the normal vector (a, b, c). Therefore, two planes with the same sign for d will be on the same side of the origin, while two planes with different signs for d will be on opposite sides of the origin. This is true for any pair of planes, whether they are parallel or not. Parallel planes are simply a special case where the normal vectors are parallel, but the same principle applies to all planes.
 

1. What are parallel planes?

Parallel planes are two-dimensional surfaces that are always the same distance apart and never intersect.

2. How do you determine if two planes are parallel?

To determine if two planes are parallel, you can use the following criteria:

  • Their normal vectors are parallel (i.e. have the same direction)
  • Their distance from the origin is the same
  • Their equations have the same slope and y-intercept

3. Can parallel planes have the same or opposite sides of the origin?

Yes, parallel planes can have either the same or opposite sides of the origin. This is determined by the direction of their normal vectors. If the normal vectors point in the same direction, the planes will be on the same side of the origin. If the normal vectors point in opposite directions, the planes will be on opposite sides of the origin.

4. What is the significance of parallel planes in geometry and physics?

In geometry, parallel planes are important for understanding concepts such as distance, angles, and symmetry. In physics, parallel planes are used to represent uniform electric fields, gravitational fields, and other physical phenomena.

5. Can parallel planes intersect at any point?

No, parallel planes can never intersect at any point. As mentioned earlier, parallel planes are always the same distance apart and never intersect. If two planes were to intersect, they would no longer be considered parallel.

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