Answer
$P(1 / 6,1 / 3,1 / 6)$ is the closest point to the origin that is on the given plane.
Work Step by Step
Taking into account that the square of the distance is given by
\[
d^{2}(P, 0)=f(x, y, z)=x^{2}+y^{2}+z^{2}
\]
We can minimize $f$ subject to the constraint $1=x+2 y+z$ using Lagrange Multipliers.
\[
\nabla f=\lambda \nabla g \Rightarrow 2 x i+2 y j+2 z k=\lambda(i+2 j+k)
\]
Thus:
\[
\lambda=2 x=2 y / 2=2 z \Rightarrow x=z=y / 2
\]
Using these relationships on the constraint:
\[
x+2 y+z=1 \Rightarrow y / 2+2 y+y / 2=1 \Rightarrow 6 y=2 \Rightarrow y=1 / 3
\]
So, $P(1 / 6,1 / 3,1 / 6)$ is the closest point to the origin that is on the given plane.
