Finding Equations of a Plane with 3 Points

[kdinser]

If they give you 3 points that are in a plane and ask you to find the equation of the plane, is there a systematic way to decide how to construct the vectors?

For instance, if I have 3 points, a(0,1,1) b(1,0,1) c(1,1,0), I could use ca x cb to find the normal to the plane. I could also, just as easily, use ab x ac. The problem is, it makes it tough to check my answers with the book if I don't pick the same ones they do.

Is there some standard way for doing this? Thanks.

 

[siddharth]

Here is one way
Let the general equation of the plane be
$$Ax + By + Cz + d =0$$
Now the plane passes through 3 points. So substitute the points in the above equation. You will have three equations. Solve for A,B and C in terms of d. Substitute them back and you will get your required equation

 

[TD]

Not really, perhaps your book prefers to use the same vectors every time but as you say, you can do it several ways and they should all give the same.

Another way to get the carthesian equation using lineair algebra would be expanding the following determinant to the first row.

$$\left| {\begin{array}{*{20}c}<br /> x & y & z & 1 \\<br /> {x_1 } & {y_1 } & {z_1 } & 1 \\<br /> {x_2 } & {y_2 } & {z_2 } & 1 \\<br /> {x_3 } & {y_3 } & {z_3 } & 1 \\<br /> \end{array}} \right| = 0$$

In this determinant, you can also replace points by vectors giving the direction of the plane, the 1 in the last column then becomes a 0. (This is easy to see since the difference of 2 points gives such a vector)

 

[HallsofIvy]

In any case, you shouldn't "check your answers" by comparing them to the books answers- If your equation is correct, then the coordinates of the three points will all satisfy the equation!