The lines \(\vec{r} = (\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(2\hat{\imath}+\hat{\jmath}-\hat{k})\) and \(\vec{r} = (3\hat{\imath}+\hat{\jmath}+\hat{k})+\mu(-4\hat{\imath}-2\hat{\jmath}+2\hat{k})\) are
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → parallel and distinct lines
The direction vectors are \(2\hat{\imath}+\hat{\jmath}-\hat{k}\) and \(-4\hat{\imath}-2\hat{\jmath}+2\hat{k}\).
\(-4\hat{\imath}-2\hat{\jmath}+2\hat{k} = -2(2\hat{\imath}+\hat{\jmath}-\hat{k})\)
So the direction ratios are proportional and the lines are parallel. Parallel lines can never be skew and can meet only if they are the same line.
The second line passes through (3, 1, 1). A point of the first line is \((1+2\lambda,\ -2+\lambda,\ 3-\lambda)\):
\(1+2\lambda = 3\) gives \(\lambda = 1\)
then \(-2+\lambda = -1 \ne 1\)
So (3, 1, 1) is not on the first line, and the lines are parallel and distinct.