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### Point on a Plane

```
Date: 06/13/97 at 19:53:42
From: Brandon Schnell
Subject: Finding a point on a plane

How can I find a point on a plane? For example, if I know (x1,y1,z1),
(x2,y2,z2) and (x3,y3,z3) define a plane, how could I solve for the
unknown z of a point assuming I know the x and y coordinates?

For example:

(B)__________ (C)
|         /
|      * /
|      D/
|      /
|     /
|    /
|   /
|  /
| /
|/
(A)

A = (1,1,1)
B = (1,10,10)
C = (5,10,5)
D = (2,9,x)

Any help would be appreciated!
```

```
Date: 06/14/97 at 08:15:40
From: Doctor Anthony
Subject: Re: Finding a point on a plane

You can use the three given points to derive the equation of the
plane, then simply substitute the values x = 2, y = 9 into this
equation to find z.

Suppose the equation of the plane is: a(x-1) + b(y-1) + c(z-1) = 0 (1)

Then (1,10,10) is on the plane: a(0) + b(9) + c(9) = 0   (2)

Also (5,10,5) is on the plane: a(4) + b(9) + c(4) = 0    (3)

Eliminating a,b,c from (1), (2) and (3) gives the determinant:

| x-1    y-1     z-1 |
|  0      9       9  |   = 0
|  4      9       4  |

So (x-1)(-45) - (y-1)(-36) + (z-1)(-36) = 0

Dividing through by -9:

5(x-1) - 4(y-1) + 4(z-1) = 0
5x - 4y + 4z - 5 = 0    (4)

Now if x = 2, y = 9, equation (4) gives:

10 - 36 + 4z - 5 = 0
4z = 36 - 10 + 5
4z = 31
z = 31/4

Thus the coordinates of your third point are (2,9, 31/4).

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry

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