anonymous
  • anonymous
What is the significance of the constant in a plane equation?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
I read somewhere that it is the distance to the origin. Is this correct? Also distance to what? The center of the plane?
phi
  • phi
Do you know about vectors and the dot product between 2 vectors ?
anonymous
  • anonymous
yes this is what I am learning. not quite sure I have a full grasp on it just yet

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phi
  • phi
Let me try again. let u be a vector with length 1, |u|= 1 \[ v \cdot u = |v| |u| \cos \theta = |v| \cos \theta \] at θ = 0 you get |v| cos 0 = |v| |dw:1371566156694:dw|
phi
  • phi
let's say the length of vector v in the above picture is K now consider another vector w, where \[ w \cdot u = K \] |dw:1371566296257:dw|
phi
  • phi
notice that the "head" of w "points" to a specific spot. If we look at all w vectors where u dot w = K, the "spots" or points described by the "head" of all vectors w will describe a line in 2D or a plane in 3D
phi
  • phi
maybe this will help http://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx
anonymous
  • anonymous
So in something like x +2y + 3z = 5, what does the 5 tell me?
anonymous
  • anonymous
the coefficients tell me the information for the normal vector to the plane, but i was just curious what the 5 means?
phi
  • phi
the normal vector to the plane is (1,2,3) its magnitude is sqrt(1+4+9)= sqrt(14) let P= (x,y,z) so that (1,2,3) dot P =5 means 1*x + 2*y + 3*z = 5 i.e. the equation of a plane if we normalize the normal vector by its magnitude (1,2,3)/sqrt(14) dot P = 5/sqrt(14) P dot with a unit vector gives the "projection" of P onto the normal It is the length of P in the direction of the normal 5/sqrt(14) tells you the distance from the origin (0,0,0) to the nearest point on the plane from the origin is 5/sqrt(14)
phi
  • phi
|dw:1371577727255:dw|
phi
  • phi
so the 5 by itself does not tell you much. You must divide it by the length of the normal, and that tells you the distance between the origin and the plane (defined to be the shortest distance between the plane and the origin)

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