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Give an example of a straight line $l$ in $\mathbb R^3$, given by a system of two equations, and a point $(a,b,c)\in \mathbb R^3$ such that there are infinitely many planes in $\mathbb R^3$ passing through $l$ and $(a,b,c)$. Justify your answer.

Straight line defined by $x−y=1,y−z=2$ and point $(1,2,3)$ (from a previous question) rather than point $(1,2,3)$ can we just give any point on the line because as long as its on the straight line there will be infinitely many planes?

Is this correct logic? its a strange question, I have never attempted one like this before so sorry if the answer is trivial.

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That point isn't on that line (just plug it in).

Other than that, yes, you could take any point on the line.

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  • $\begingroup$ So could I just state any point on the line with an explanation centred around the fact that you'll get infinitely many planes if and only if they lie on a straight line. $\endgroup$ – Daniel Jul 14 '15 at 15:26
  • $\begingroup$ @Jack: Well, the "only if" part is true, but you don't need it; they just asked for any such point and line. So it's enough to argue that there are infinitely many planes passing through a given line, and that if the point is on the line then they all also pass through the point. $\endgroup$ – joriki Jul 14 '15 at 15:28
  • $\begingroup$ This seems right to me, the question was on an old test paper and is worth 6/60...so I'd presume that explanation would be proficient $\endgroup$ – Daniel Jul 14 '15 at 15:33
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can we just give any point on the line because as long as its on the straight line there will be infinitely many planes?

I'm not sure if this is enough of a justification or rigorous enough. It merely states that "the solution is correct, because it fullfills all requirements of the task", but it doesn't show how that's the case.

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  • $\begingroup$ I must admit I do find the question a bit odd, if its not a case of stating a point on the line what else could be done? $\endgroup$ – Daniel Jul 14 '15 at 15:25

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