Statement 1: The shortest distance between the lines x2= y-1= z2 and x-14=y-1-2=z-14 is √2…

Question

Statement 1: The shortest distance between the lines $\dfrac x2=\dfrac y{-1}=\dfrac z2$ and $\dfrac{x-1}4=\dfrac{y-1}{-2}=\dfrac{z-1}4$ is $\sqrt2$.
Statement 2: The shortest distance between two parallel lines is the perpendicular distance from any point on one of the lines to the other line.

Accepted Answer

Both lines have direction $(2,-1,2)$ (the second line's direction $(4,-2,4)=2(2,-1,2)$), so the lines are parallel --- Statement 2 is the standard, correct fact.
Using points $O=(0,0,0)$ on line 1 and $A=(1,1,1)$ on line 2: $\overrightarrow{AO}=(-1,-1,-1)$,
$$\overrightarrow{AO}\times(2,-1,2)=(-3,0,3),\qquad |{-3,0,3}|=3\sqrt2,\qquad |(2,-1,2)|=3.$$
$$\text{Distance}=\frac{3\sqrt2}3=\sqrt2,$$
confirming Statement 1 is true, and Statement 2 correctly explains it (this is the standard method for parallel lines).

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