Q.By computing the shortest distance, determine whether following lines intersect each other. $\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j}) + \mu(\hat{i} + \hat{j} - \hat{k})$

[!TLDR] Compute $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)$ for the two given lines; it is nonzero, so they do not intersect.

With $\vec a_1=\hat i-\hat j$, $\vec b_1=2\hat i+\hat k$, $\vec a_2=2\hat i-\hat j$, $\vec b_2=\hat i+\hat j-\hat k$: the scalar triple produ …

Here $\vec a_1=\hat i-\hat j$, $\vec b_1=2\hat i+\hat k$, $\vec a_2=2\hat i-\hat j$, $\vec b_2=\hat i+\hat j-\hat k$.

$(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=1(-1)+0(3)+0(2)=-1\ne0$ …

Apply the intersection test $(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=0$; a nonzero value means the li …

Concluding the lines are parallel just because the triple product is nonzero -- nonzero only rules out intersection, a separate check (direction ratios not proportional) is needed to confir …