What you’ll learn
Use precise point/line/plane terminology.
Key idea
Plane uniqueness requires an off-line third point; a line alone can lie in multiple planes.
Two distinct points fix a line, not a plane; a suitable third point must lie off that line.
Worked example
Enter the exact value or ordered values in the format requested below.
Coordinate configuration in 3D: A=(-2, -2, 0), B=(2, -2, 0), C=(-2, 0, 0); AB is horizontal and C is off line AB. A later construction starts from 2 fixed points; for now, how many distinct planes contain ALL three displayed points?
- A and B are distinct since their first coordinates differ by 4; C is off AB since its second coordinate differs by 2.
- Three noncollinear points determine exactly one plane. Starting with 2 suitable fixed points requires 1 suitable additions; a line alone does not fix the plane.
Answer: 1
Quick Check
Evaluate a worked attempt at use precise point/line/plane terminology, identify the first invalid step, and provide the correction.
Coordinate configuration in 3D: A=(-2, 0, 0), B=(0, 0, 0), C=(-2, 3, 0); AB is horizontal and C is off line AB. A construction will start with 1 fixed points. Student: Step 1, A and B determine a unique line. Step 2, that line alone determines a unique plane even before C is specified. Identify the first invalid step and repair.