How Do You Find the Equation of a Plane Equidistant from Two Points?

[Nikitin]

Homework Statement

A plane lies inbetween two points A=(2,0,2) and B=(4,2,0). Let a point P=(x,y,z) be equally far away from A as from B.

Find the equation for this plane (like in this format: [PLAIN]http://upload.wikimedia.org/math/6/7/8/67834a730a04bb1f3d6ceab80f5284e4.png )

The Attempt at a Solution

Well.. What can I say. I basically got two points and I'm supposed to make a plane out of it (lmao), with no directional vectors or anything.

I also thought that a plane was infinitely big so I don't understand how a plane can be inbetween two points as explained in the assignment... The plane contains the two points is more correct to say, right?

Anyway can't you make an infinite number of planes with the given information? While a plane's equation (like this: [PLAIN]http://upload.wikimedia.org/math/6/7/8/67834a730a04bb1f3d6ceab80f5284e4.png ) describes one specific plane with a specific starting point and orientation?

 

[Hootenanny]

Homework Statement

A plane lies inbetween two points A=(2,0,2) and B=(4,2,0). Let a point P=(x,y,z) be equally far away from A as from B.

Find the equation for this plane (like in this format: [PLAIN]http://upload.wikimedia.org/math/6/7/8/67834a730a04bb1f3d6ceab80f5284e4.png )

The Attempt at a Solution

Well.. What can I say. I basically got two points and I'm supposed to make a plane out of it (lmao), with no directional vectors or anything.

I also thought that a plane was infinitely big so I don't understand how a plane can be inbetween two points as explained in the assignment... The plane contains the two points is more correct to say, right?

Anyway can't you make an infinite number of planes with the given information? While a plane's equation (like this: [PLAIN]http://upload.wikimedia.org/math/6/7/8/67834a730a04bb1f3d6ceab80f5284e4.png ) describes one specific plane with a specific starting point and orientation?

Yes you can make a single plane with this information. No, the plane does not contain the two points. The key point to note is that the two points are equidistant from all points on the plane.

To help you visualize this, consider a table top. Place one golf ball a distance 'd' above the table and a second ball a distance 'd' underneath the table, directly below the first. The two golf balls are your points and the table is your plane. Notice that if you tilt the table, but leave the golf balls in the same position, they will no longer be equidistant form all points on the plane. This is the key point. The two points are equidistant from all points on the plane.

Now, returning to your problem. Can you link the line joining the two points to the normal of the plane?

 

[Nikitin]

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOH! Now I get it

I thought the points were inbetween the plane as, in, like the golfballs were at two sides of the tabletop.

The directional vector would be AB. Wouldn't a point on the plane thus be A+0,5AB = (2,0,2)+0,5[2,2,-2]=(2,0,2) + [1,1,-1]=[3,1,1]

Then the equation 4 this stuff would simply be [2,2,-2]*[X-3,Y-1,Z-1]=
2x-6+2Y-2-2z+2 = 2x+2y+2z-6 = 0 ==> x+y+z-3=0

thanks 4 the help :)

 

[Hootenanny]

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOH! Now I get it

I thought the points were inbetween the plane as, in, like the golfballs were at two sides of the tabletop.

The directional vector would be AB. Wouldn't a point on the plane thus be A+0,5AB = (2,0,2)+0,5[2,2,-2]=(2,0,2) + [1,1,-1]=[3,1,1]

Then the equation 4 this stuff would simply be [2,2,-2]*[X-3,Y-1,Z-1]=
2x-6+2Y-2-2z+2 = 2x+2y+2z-6 = 0 ==> x+y+z-3=0

thanks 4 the help :)

Looks good to me