Do these three points lie in the same plane $\mathbb{R}^{3}$?
$(1,0,0), (1,0,0), (0,0,0)$
I read that they lie in the same plane when they are different? But It wasn't said how different they need to be. All of them need to be different from one another? In this example I wouldn't be sure because only two of them are not different. But I would say they do not lie in the same plane.
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$\begingroup$Any plane though the $x$ axis includes both of the two points. The fact that you list one point twice does not change the fact that there are only two distinct points in the list.
$\endgroup$ $\begingroup$Any two distinct points lie on a unique line. There are an infinity of planes containing this line.
Any three distinct non-collinear points lie an a unique plane.
Distinct means different from each other. You list three points but two are the same so you only have two distinct points and thus one line, an infinity of planes.
Think of gluing a stack to a flat piece of cardboard. The stick is your line and the cardboard is your plane. If you hold the ends of the stick in place you can turn the plane around to an infinity of positions.
Non-collinear means not all in the same straight line. You can test this with vectors. Vector AB = B - A, subtracting the corresponding coordinates. Vector AC = C - A. If AC is a scalar multiple of AB, then the points are collinear so you have a unique line but again an infinity of planes.
Unique means one and only one possible. Again, two distinct points define a unique line through them. Three distinct non-collinear points define a unique plane through them.
Your statement of your problem is vague and confusing. You talk about if "they" are the same or if "they" are different. It is not clear who you mean by "they", two of the points or all three of them or whatever. For your own improved undertanding, learn and use the above vocabulary. It really helps.
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