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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 $x$ $+2 y+3 z=1$. 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 $x+2 y+3 z=1$ is achieved when the point ( $x, y$, z) is the base 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

## 1

$1 / 14$