Two parallel cables run along the lines r = (î + ĵ + k̂) + λ(î + 2ĵ + 2k̂) and r = (3î - 7ĵ - k̂) +…

Two parallel cables run along the lines $\vec{r} = (\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = (3\hat{i} - 7\hat{j} - \hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$, with lengths in metres. A student finds the distance between the two points $\vec{a}_1$ and $\vec{a}_2$ and says it is the distance between the cables. Find the length of $(\vec{a}_2 - \vec{a}_1) \times \vec{b}$, the true distance between the cables, and the distance between the two points the student used.