Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ...

**s3a** · January 10th 2012, 06:58 AM

Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

Answer:
x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!
Thanks in advance!

**alexmahone** · January 10th 2012, 07:10 AM

Originally Posted by s3a

Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

Answer:
x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!
Thanks in advance!

You need to minimize $x^2+y^2+z^2$ .

Using the method of Lagrange multipliers,

$2x=\lambda$

$2y=3\lambda$

$2z=-\lambda$

Substituting for $x,\ y,\ z$ in the equation of the plane,

$-\lambda=\lambda+9\lambda+6$

$11\lambda=-6$

$\lambda=-\frac{6}{11}$

$x=-\frac{3}{11}=-0.272727272727273$

$y=-\frac{9}{11}=-0.818181818181818$

$z=\frac{3}{11}=0.272727272727273$

**chisigma** · January 10th 2012, 07:13 AM

Originally Posted by s3a

Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

Answer:
x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!
Thanks in advance!

Search the minimum of the function...

$f(x,y)= x^{2} + y^{2} + z^{2}= x^{2}+y^{2}+ (x+3 y+3)^{2}$

... respect to x and y...

Kind regards

$\chi$ $\sigma$

**s3a** · January 10th 2012, 07:22 AM

Sorry for the stupid question but how do I know that f(x,y) = x^2 + y^2 + z^2 is what I want to minimize?

**alexmahone** · January 10th 2012, 07:36 AM

Originally Posted by s3a

Sorry for the stupid question but how do I know that f(x,y) = x^2 + y^2 + z^2 is what I want to minimize?

Because $x^2 + y^2 + z^2$ is the square of the distance of the point $(x,\ y,\ z)$ from the origin.

**Plato** · January 13th 2012, 03:01 PM

Originally Posted by s3a

Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.
Answer:
x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

One does not need any calculus to solve this.
The plane is $x+3y-z=-3$ .
Find out where the line $<t,3t,-t>$ intersects that plane.

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Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

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