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

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

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

x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!

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

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.

x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!
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,

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

$11\lambda=-6$

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

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

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

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

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

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.

x = -0.272727272727273
y = -0.818181818181818
z = 0.272727272727273

Please show me how to do this problem!

Any help would be appreciated!
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

$\chi$ $\sigma$

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

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?

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

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.

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

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.
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 $$ intersects that plane.