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EEWannabe
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URGENT - Lagrange Multipliers
Using the method of lagrange multipliers prove the formula for the distance from a point (a,b,c) to a plane Ax + By + Cz = D
Using the equation of the form;
H(x,y,z,L) = (x-a)^2 + (y -b)^2 +(z-c)^2 + L(Ax + By + Cz - D)
Therefore by differentiating, a relationship between x, y and z can be found with lamba, nameley;
x = (-LA/2) + a
y = (-LB/2) + b
z = (-LC/2) + c
By inserting these back into the equation for a plane I get to this stage for L;
[tex] L = \frac{2(Aa + Bb + Cc - D)}{A^2 + B^2 + C^2} [/tex]
Thus, putting this back into the Lagrange equation the value for D^2 I get is (D^2 =(x-a)^2 + (y -b)^2 +(z-c)^2);
[tex]D^2 to plane = \frac {2(Aa + Bb + Cc - D)(Ax + By + Cz - D)}{A^2 + B^2 + C^2} [/tex]
However this is at ends with;
[tex] D to plane = \frac{(Aa + Bb + Cc - D)}{(A^2 + B^2 + C^2)^_\frac{1/2}} [/tex]
I can't see what's gone wrong , I know i can't expect anyone to do it straight away or anything but I have a big exam tomorrow and there'll definatley be a question on something like this, any hints would be fantastic!
Thanks again
Homework Statement
Using the method of lagrange multipliers prove the formula for the distance from a point (a,b,c) to a plane Ax + By + Cz = D
The Attempt at a Solution
Using the equation of the form;
H(x,y,z,L) = (x-a)^2 + (y -b)^2 +(z-c)^2 + L(Ax + By + Cz - D)
Therefore by differentiating, a relationship between x, y and z can be found with lamba, nameley;
x = (-LA/2) + a
y = (-LB/2) + b
z = (-LC/2) + c
By inserting these back into the equation for a plane I get to this stage for L;
[tex] L = \frac{2(Aa + Bb + Cc - D)}{A^2 + B^2 + C^2} [/tex]
Thus, putting this back into the Lagrange equation the value for D^2 I get is (D^2 =(x-a)^2 + (y -b)^2 +(z-c)^2);
[tex]D^2 to plane = \frac {2(Aa + Bb + Cc - D)(Ax + By + Cz - D)}{A^2 + B^2 + C^2} [/tex]
However this is at ends with;
[tex] D to plane = \frac{(Aa + Bb + Cc - D)}{(A^2 + B^2 + C^2)^_\frac{1/2}} [/tex]
I can't see what's gone wrong , I know i can't expect anyone to do it straight away or anything but I have a big exam tomorrow and there'll definatley be a question on something like this, any hints would be fantastic!
Thanks again
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