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# Engineering

The lines \overrightarrow{r}=(\widehat{i}-\widehat{j})+l(2\widehat{i}+\widehat{k}) and \overrightarrow{r}=(2\widehat{i}-\widehat{j})+m(\widehat{i}+\widehat{j}-\widehat{k})
Option: 1 do not intersect for any values of l and m
Option: 2 intersect for all values of l and m
Option: 3 intersect for the values when l=2 and m=\frac{1}{2}
Option: 4 intersect for the values when l=2 and m=2

Answers (1)
S Suraj Bhandari

\overrightarrow{r}=(\widehat{i}-\widehat{j})+l(2\widehat{i}+\widehat{k})

=(2l+1)\widehat{i}-\widehat{j}+l\;\widehat{k}

\overrightarrow{r}=(2\widehat{i}-\widehat{j})+m(\widehat{i}+\widehat{j}-\widehat{k})

=(m+2)\widehat{i}+(m-1)\widehat{j}-m\widehat{k}

For intersection

\\1+2 \ell=2+\mathrm{m}\qquad\ldots(i)\\ -1=m-1\qquad\quad\ldots(ii)\\ \ell=-\mathrm{m}\qquad\qquad\quad\ldots(iii)\\ \text {from (ii) } m=0

\text {from (iii) } \ell=0

These values of m and l do not satisfy equation (1). Hence the two lines do not intersect for any values of l and m

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