 Solution

The common ratio supplies a line parameter $s$: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 $s_{\ast }=-9/29$ and the foot is $(11,2,-7)/29$. ThusThe cross product formula $d=|\mathbf{p}\times \mathbf{v}|/|\mathbf{v}|$ agrees, since $\mathbf{p}\times \mathbf{v}=(1,-2,1)$.