Two lines \(L_1\) and \(L_2\) in space are both perpendicular to a line \(L_3\). Then \(L_1\) and \(L_2\)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → may be parallel, intersecting or skew
In a plane, two lines perpendicular to the same line are parallel. In space this is no longer true. Take \(L_3\) to be the z-axis; every line with direction ratios a, b, 0 is perpendicular to it.
\(L_1\) = the x-axis and \(L_2\) = the line \(\vec{r} = \hat{k}+\mu\hat{\imath}\): parallel
\(L_1\) = the x-axis and \(L_2\) = the y-axis: they intersect at the origin, at right angles
\(L_1\) = the x-axis and \(L_2\) = the line \(\vec{r} = \hat{k}+\mu\hat{\jmath}\): skew, since y = z = 0 on \(L_1\) but z = 1 on \(L_2\)
The first example also shows the lines need not intersect or be perpendicular, and the second shows they need not be parallel, so options (1), (2) and (3) all fail.