Equation of plane through points on x,y,z axes

[iomtt6076]

Homework Statement

Find the equation of the plane through the points on the x, y, and z-axes where x=a, y=b, z=c, respectively.

Homework Equations

If <A,B,C> is a vector perpendicular to the plane, then A(x-a)+B(y-b)+C(z-c)=0 is the equation of the plane.

The Attempt at a Solution

I know this is extremely simple, but I can't get the right answer even using different methods.

Method 1:
The vectors <0,b,-c> and <a,-b,0> are in the plane. The cross product <0,b,-c> X <a,-b,0> gives <-bc,-ac,-ab>, which is normal to the plane. Then the equation of the plane is (-bc)(x-a)+(-ac)(y-b)+(-ab)(z-c)=0, and simplifying I get (x/a)+(y/b)+(z/c)=3. Wrong.

Method 2:
Substituting the 3 points (a,0,0), (0,b,0), (0,0,c) into A(x-a)+B(y-b)+C(z-c)=0 gives

-Bb-Cc=0
-Aa-Cc=0
-Aa-Bb=0.

Thus C=-Aa/c, B=-Aa/b and substituting this back into A(x-a)+B(y-b)+C(z-c)=0 and simplifying I get (-x/a)+(y/b)+(z/c)=1, which is different than the answer from Method 1 and still wrong.

 

[tiny-tim]

Hi iomtt6076!

Find the equation of the plane through the points on the x, y, and z-axes where x=a, y=b, z=c, respectively.

eugh … you've made it so complicated.

Just call the plane px + qy + rz = 1, and solve! ​

 

[iomtt6076]

tiny-tim, thank you for your reply. I see how to get from px + qy + rz = 1 to the correct answer and at last realize my (embarrassing) mistake.

In Method 1, it was incorrect to say that "the equation of the plane is (-bc)(x-a)+(-ac)(y-b)+(-ab)(z-c)=0." This equation comes from setting the dot product <-bc,-ac,-ab>*<x-a,y-b,z-c> equal to 0. But (a,b,c) is NOT on the plane. It should be <-bc,-ac,-ab>*<x-a,y,z>.

In Method 2, my mistake can be traced to my statement under "

Homework Equations

": A(x-a)+B(y-b)+C(z-c)=0 is NOT the equation of the plane because again, (a,b,c) is not on the plane.