1. Write each line in component form

The given lines are

$\vec r=(\hat i-\hat j)+l(2\hat i+\hat k)$ $\vec r=(2\hat i-\hat j)+m(\hat i+\hat j+\hat k)$

So their parametric forms are:

For line 1: $\vec r=(1,-1,0)+l(2,0,1)$ Hence, $x=1+2l,\quad y=-1,\quad z=l$

For line 2: $\vec r=(2,-1,0)+m(1,1,1)$ Hence, $x=2+m,\quad y=-1+m,\quad z=m$

2. Condition for intersection

For the two lines to intersect, there must exist values of $l$ and $m$ such that corresponding coordinates are equal:

$1+2l=2+m \quad ...(1)$ $-1=-1+m \quad ...(2)$ $l=m \quad ...(3)$

3. Solve the system

From equation (2): $-1=-1+m \Rightarrow m=0$

From equation (3): $l=m=0$

Now check in equation (1): $1+2(0)=2+0$ $1=2$

This is false.

So no pair $(l,m)$ satisfies all three equations simultaneously.

4. Conclusion

The two lines do not intersect for any values of $l$ and $m$.

Therefore, the correct option is:

$\boxed{\text{A}}$

5. Check the given options

• A: do not intersect for any values of $l$ and $m$ ✅
• B: intersect for all values of $l$ and $m$ ❌
• C: intersect when $l=2, m=\tfrac12$ ❌
• D: intersect when $l=1, m=2$ ❌