= Solution
The common ratio supplies a <line> parameter $s$:
$$
\boxed{\mathbf r=\mathbf p+s\mathbf v,\qquad\mathbf p=(1,1,1),\quad\mathbf v=(2,3,4).}
$$
The nearest point is the <orthogonal projection> of the origin onto the <line>. Minimize $|\mathbf p+s\mathbf v|^2$: its <derivative> is $2\mathbf v\cdot(\mathbf p+s\mathbf v)$, so $s_*=-9/29$ and the foot is $(11,2,-7)/29$. Thus
$$
\boxed{d=\sqrt{\frac6{29}}.}
$$
The <cross product> formula $d=|\mathbf p\times\mathbf v|/|\mathbf v|$ agrees, since $\mathbf p\times\mathbf v=(1,-2,1)$.
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