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No, because you need at least three points to define a plane.
Look at a corner in the room you're in, where two different walls meet. Take a pencil and mark two points on the corner. If you think of the walls as two planes, then you have two different planes that contain the two points you marked. Therefore, you can't have 2 points define a plane.
Similarly, look up at the ceiling where two walls and the ceiling meet in the upper corner. That's a point common to THREE different planes. So 1 point doesn't determine a specific plane. 
Yes. One point is on the plane, the second point is above it defining a vector (line) perpendicular to the plane.
Many people falsely assume the points must be on the plane (like a camera tripod) so it must take three. The two point analogy is more like a nail thought a board with the two points at each end of the nail. 
Two points can never determine a plane ... they can only form a straight line as in "two points determine a straight line."
And, of course, a point is just a point. It cannot be a line nor can it be a plane. However, a point can be on a line and on a plane. 
They don't. Two points detremine a line, and there are an infinite number of planes which include the line.

I can show it but with a assumption that they don't lie on a line passing through origin :P. This is bit of cheating but its interesting. consider two different points with coordinates (x1, y1, z1) and (x2, y2, z2). and they are linearly independent. then they can span a plane
Can you show that 2 Points Determine a Plane?
We have all heard that "3 Points Determine a Plane"
Meaning 3 Distinct NonCollinear Points in 3Space
Using the same ideas  Can you show that:
2 distinct Points Determine a Plane
or better
1 Point Determines a Plane
Meaning 3 Distinct NonCollinear Points in 3Space
Using the same ideas  Can you show that:
2 distinct Points Determine a Plane
or better
1 Point Determines a Plane
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