Find the value of $p$ if the shortest distance between the lines
\[
\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \quad \text{and} \quad \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})
\]
is exactly $\frac{3}{\sqrt{2}}$ units.
Show Hint
An absolute value equation $|x| = c$ always splits into two cases: $x = c$ and $x = -c$. Forgetting the negative case means losing half of the valid solutions!