What information is needed to uniquely determine a plane?

Question

BlackTom AI · Accepted Answer

A plane in R^3 can be written n · (r − r0) = 0, where n is a nonzero normal vector and r0 is the position vector of a specific point on the plane.

1. Relevant knowledge points and concepts
- A plane’s orientation is fixed by a normal vector n (a vector perpendicular to every direction in the plane).
- A plane’s location (which parallel plane it is) is fixed by any single point r0 that lies on it.
- The standard vector equation of a plane: n · (r − r0) = 0 (equivalently n · r = n · r0).

2. Complete reasoning process step by step
- To uniquely specify a plane you must fix both its orientation and its position.
- Orientation: the normal vector n gives the unique direction perpendicular to the plane; any vector orthogonal to n lies in the plane.
- Position: without a point r0, the equation n · r = c has c unspecified; different values of c give parallel planes with the same normal. Thus a point (which yields c = n · r0) is required to pick the specific plane.
- Combine the two into the plane equation n · (r − r0) = 0; this equation determines exactly the set of points r that lie on the plane.

3. Intermediate steps (algebraic)
- Given n and r0, the plane is { r | n · (r − r0) = 0 }.
- If r = r0 + s v1 + t v2 for two independent in-plane direction vectors v1, v2, then n · v1 = n · v2 = 0 and the equation holds.

4. Why the correct answer is correct
- A normal vector fixes orientation; a point fixes the plane’s location. Together they give one and only one plane, as shown by the unique scalar c = n · r0 in n · r = c.

5. Why the other options are incorrect
- "A point on the plane and a direction vector.": This gives a point and one direction through it — that defines a line, not a unique plane (infinitely many planes contain that line).
- "Two direction vectors.": Two independent direction vectors determine an oriented plane through the origin (their span) but without a point you cannot locate the plane in space (you need a position/translation). If they are dependent, they do not determine a plane at all.
- "A normal vector and a direction vector.": These give orientation information but no position; also a direction vector lying in the plane must be orthogonal to the normal, so this pair is redundant for orientation and still leaves infinitely many parallel planes.

6. Conclusion
Therefore the unique specification requires a point on the plane and a normal vector, matching the provided answer: "A point on the plane and a normal vector."

What information is needed to uniquely determine a plane?单项选择题

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