Cartesian scalar equation of plane

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  • #1
yxj
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Main Question or Discussion Point

Just wanted to confirm. Cartesian scalar equation of plane refers to equation of plane right?
As in Ax+By+Cz=D. which i think is the vector equation of a plane. I'm getting confused and need clarification
thank you

edit= ok sorry.. i think i got it figured out =p scalar is there because.. equation of plane is a dot product. right?
 
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Answers and Replies

  • #3
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The expression

ax+by+cz+d=0 refers to a plane and is the general scalar Cartesion expression.

(Aside I wonder if the other expression you were referring to is the direction cosine version

lx+my+nz+p=0.)

This in itself is not a vector expression either, but it does lead to the identification of a unique vector, normal to the plane.

This vector, n = (a,b,c)

The vector expression for a plane is given by

[tex]n.(r - {r_0})[/tex]

edit should read

[tex]n.(r - {r_0})=0[/tex]

n intersects the plane at the point [tex]{r_0} = ({x_0},{y_0},{z_0})[/tex]

r is the position vector [tex]r = (x,y,z)[/tex]


ax+by+cz
This part of the equation defines a series of parallel planes all normal to the vector n
d
selects the particular plane of interest
 
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  • #4
yxj
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and cartesian form would be (x,y,z).(#,#,#)=# or Ax+By+Cz=D
 
  • #5
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Cartesian refers to rectangular coordinates x,y,z.
As opposed to some other coordinate system eg r,[tex]\phi[/tex],[tex]\theta[/tex]
 
  • #6
yxj
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erm.. thah means.. if they were to ask cartesian form of equation of plane should i write
a) (x,y,z).(A,B,C)=D aka r.(A,B,C,)=D
b) Ax+By+Cz=D
 
  • #7
5,439
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ax+by+cz+d=0 refers to a plane and is the general scalar Cartesion expression.
I've said it once.

You can put the d on the other side of the equation if you like, so long as you are careful to get the signs right.


With regards to your last post


Both (a) and (b) are cartesian since n and r are cartesian vectors.

Cartesian refers to the coordinate system, not the vectors or the planes themselves.


(b) I think (b) is the form you are looking for.

(a) is not quite correct - the expression should not contain d - this is already included in [tex]{r_0}[/tex] - your expression should equal zero, not =D.

Sorry I missed the =0 from the vector expression earlier I have amended that post.
 
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  • #8
yxj
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thank you so much.. it's much clearer now xD
 
  • #9
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Glad to help.
 

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