Find Perpendicular Planes Intersecting at a Point

In summary, to find the equations of two perpendicular planes intersecting at a point (1,2,5) with the plane 3x+y+z=10, you can use the following equations: Ax+By+Cz+D=0, Ex+Fy+Gz+H=0, and Ix+Jy+Kz+W=0, where the normal vectors (A,B,C), (E,F,G), and (I,J,K) are perpendicular to each other. The intersection point must satisfy all three equations, so if it is (x_1,y_1,z_1), then Ax_1+By_1+Cz_1+D=0, Ex_1+Fy_1
  • #1
mathman99
3
0
How to find (equations of) two perpendicular planes intersecting to a plane (say 3x+y+z=10) in a point (say 1,2,5 ).
All the three planes are perpendicular to each other and intersecting at a single point (say (1,2,5) in this example)
If possible explain it in vector form and non-vector forms.

Thanks in advance.
 
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  • #2
So you got three perpendicular planes:

Ax+By+Cz+D=0
Ex+Fy+Gz+H=0
Ix+Jy+Kz+W=0

If two planes are perpendicular to each other then, then their normal vectors are also perpendicular to each other.

The normal vectors are (A,B,C), (E,F,G) and (I,J,K). This means that the scalar product (A,B,C) o (E,F,G) = 0 , (A,B,C) o (I,J,K) =0 , (E,F,G) o (I,J,K)=0

Also the vector product of two of them gives us the third vector.
For ex. (A,B,C) x (E,F,G) = (I,J,K)

The intersection point is standing on all of the planes. So that if it is [itex](x_1,y_1,z_1)[/itex] then:
[tex]Ax_1+By_1+Cz_1+D=0 [/tex]
[tex]Ex_1+Fy_1+Gz_1+H=0 [/tex]
[tex]Ix_1+Jy_1+Kz_1+W=0[/tex]Regards.
 

1. What is the definition of perpendicular planes intersecting at a point?

Perpendicular planes intersecting at a point refer to two planes that meet at a 90-degree angle at a specific point in space.

2. How do you find the point of intersection between two perpendicular planes?

To find the point of intersection, you can set up a system of equations using the equations of the two planes and solve for the common variables. The resulting values will give you the coordinates of the point of intersection.

3. Can perpendicular planes intersect at more than one point?

No, perpendicular planes can only intersect at one point. This is because if they were to intersect at more than one point, they would not be considered perpendicular planes.

4. What is the significance of perpendicular planes intersecting at a point in geometry?

Perpendicular planes intersecting at a point have a special geometric relationship. This means that they are always at a right angle to each other, which can be useful in many geometric calculations and constructions.

5. How can you prove that two planes are perpendicular?

To prove that two planes are perpendicular, you can show that the dot product of their normal vectors is equal to zero. This means that the two planes are orthogonal, or perpendicular, to each other.

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