By computing the shortest distance determine whether the following pairs of line intersect or not: ` vec r=( hat i- hat j)+lambda(2 hat i+ hat k); vec r=2 hat i- hat j+mu( hat i- hat j- hat k)` `(x-1)/2=(y+2)/3=z ;(x+1)/5=(y-2)/1;z=2` .

Explanation

To determine whether the given pairs of lines intersect, we will compute the shortest distance between them. If the shortest distance is not equal to zero, the lines do not intersect.

### Step-by-Step Solution:

1. **Identify the Line Equations**:
The first line can be expressed as:
\[
\vec{r_1} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{k})
...