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In two lines lie in different plane, can they be parallel to each other?

I am thinking if two lines are parallel to each other, then their direction

cosines must be same so two lines lying in different plane cannot be parallel. Am I missing something? Ni

Jean Marie
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Sid1234
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  • The conclusion is correct, yes. – Semiclassical Jun 11 '16 at 05:56
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    Because a line (presumably in 3D space) lies on infinitely many different planes I would rephrase the statement to read: If two lines are parallel, then there exists a plane they both lie on. Phrased that way your conclusion is correct. The trouble I have with your phrasing is that it leaves the specification of the plane ambiguous. For example we can select a line $L_1$ on the plane $x=0$ and another $L_2$ from the plane $y=1$ in such a way that $L_1$ and $L_2$ are parallel, but "they lie on different planes". Heck, it might even happen that the two lines coincide. – Jyrki Lahtonen Jun 11 '16 at 05:58
  • First of all a) either your planes (P1) and (P2) intersect with intersection line (L) or b) they are parallel. In case a) take in (P1) and (P2) any parallel to (L) – Jean Marie Jun 11 '16 at 06:16

1 Answers1

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Example 1. In $R^3$ let $P=\{(x,y,z):y=0\}$ and $Q=\{(x,y,z):z=0\}.$ Let $L_1=\{(x,y,z):y=0\land z=1\}$ and $L_2=\{(x,y,z):z=0\land y=1\}.$ Then $L_1\in P$ and $L_2\in Q .$ And $L_1, l_2$ are parallel.

Example 2. Take a greeting-card, partly opened. Find line-segments on the front and on the back that are parallel.

DanielWainfleet
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