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Problem 589

Give an example to show that two non-parallel lines in \(\mathrm{R}^{3}\) need not intersect.

Expert verified

An example of two non-parallel lines in \(\mathrm{R}^{3}\) that do not intersect are \( L_{1}: (x, y, z) = (1, 2, 3) + t(1, -1, 0) \) and \( L_{2}: (x, y, z) = (2, 4, 1) + s(1, 1, -1) \). The direction vectors for these lines are \( \vec{a} = (1, -1, 0) \) and \( \vec{b} = (1, 1, -1) \), which are not proportional, confirming that the lines are non-parallel. Solving for parameters t and s to check for intersection points, we find that they do not have a unique intersection point, indicating that the lines do not intersect. This example shows that in \(\mathrm{R}^{3}\), two non-parallel lines do not necessarily intersect.

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