It is known that $x+2 y+3 z=1$. What is the minimum value that the expression $x^{2}+y^{2}+z^{2}$ can take?
What is the geometric interpretation of this problem

## Solution

Consider a rectangular coordinate system in space. Draw the plane P, defined by the equation $x + 2y + 3z = 1$. The expression

$x^2 + y^2 + z^2$ is equal to the square of the distance from the origin $O$ to the point with coordinates $(x, y, z)$. The distance from the origin to the point

$(x, y, z)$, lying in the plane P, does not exceed the distance from the origin to the plane P. Therefore, the minimum value of the expression $x^2 + y^2 + z^2$ under the condition $x + 2y + 3z = 1$ is achieved when the point $(x, y, z)$ is the foot of the perpendicular dropped from $O$ to the plane P, and is equal to the square of the distance from $O$ to the plane P. This distance can be calculated using a known formula. As a result, we get

$\frac{1}{1^2 + 2^2 + 3^2} = \frac{1}{14}$

## Answer