Applications to geometry

Question

Show that the distance between the two lines in the direction $(1,2,1)$ that pass through the points $(4,2,-1)$ and $(3,1,0)$, respectively is $7/ \sqrt{21}$

Answer 1

You want to minimize the distance between $(4,2,-1)$ and $(3,1,0)+(1,2,1)t$

or its square, i.e., $(-1+t)^2+(-1+2t)^2+(1+t)^2=6t^2-4t+3=6(t-\frac13)^2+\frac73$.

Can you take it from here?