# 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 $\displaystyle x^2+y^2+z^2$.

Using the method of Lagrange multipliers,

$\displaystyle 2x=\lambda$

$\displaystyle 2y=3\lambda$

$\displaystyle 2z=-\lambda$

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

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

$\displaystyle 11\lambda=-6$

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

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

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

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

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...

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

... respect to x and y...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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 $\displaystyle x^2 + y^2 + z^2$ is the square of the distance of the point $\displaystyle (x,\ y,\ z)$ 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 $\displaystyle x+3y-z=-3$.
Find out where the line $\displaystyle <t,3t,-t>$ intersects that plane.