The lines overrightarrow{r} = (hat{i} - hat{j | vedclass

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(C) The given lines are $\overrightarrow{r} = \hat{i}(1 + 2\ell) + \hat{j}(-1) + \hat{k}(\ell)$ and $\overrightarrow{r} = \hat{i}(2 + m) + \hat{j}(m -

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Do not intersect for any values of $\ell$ and $m$

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(C) The given lines are $\overrightarrow{r} = \hat{i}(1 + 2\ell) + \hat{j}(-1) + \hat{k}(\ell)$ and $\overrightarrow{r} = \hat{i}(2 + m) + \hat{j}(m - 1) + \hat{k}(-m)$.For the lines to intersect,there must exist values of $\ell$ and $m$ such that the coordinates are equal:$1 + 2\ell = 2 + m$  $(i)$$-1 = m - 1$  $(ii)$$\ell = -m$  $(iii)$From equation $(ii)$,we get $m = 0$.Substituting $m = 0$ into equation $(iii)$,we get $\ell = 0$.Now,check if these values satisfy equation $(i)$:$1 + 2(0) = 2 + 0 \implies 1 = 2$,which is a contradiction.Since the values of $\ell$ and $m$ do not satisfy all three equations simultaneously,the lines do not intersect for any values of $\ell$ and $m$.

The lines $\overrightarrow{r} = (\hat{i} - \hat{j}) + \ell(2\hat{i} + \hat{k})$ and $\overrightarrow{r} = (2\hat{i} - \hat{j}) + m(\hat{i} + \hat{j} - \hat{k})$:

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