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Points, Lines, Planes in Space Calculator

Online points/planes checker: enter three points and use the cross product to test collinearity; non-collinear points define a unique plane.

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Three non-collinear points define a plane
Point 1
x
y
z
Point 2
x
y
z
Point 3
x
y
z

📖 Tutorial | Points, Planes

1. Definition
Three non-collinear points define exactly one plane; collinear points do not. Test with whether the cross product AB×AC is the zero vector.
2. Formula & Notation
A,B,CPoints
3. Properties & Laws
  • Non-collinear points define a plane;
  • Line + off-line point, intersecting/parallel lines also do;
  • |AB×AC|=0 iff collinear;
  • Two planes meet in a line.
4. Steps
  1. Enter the three points in the groups;
  2. Click "Check";
  3. Or click "Load Example".
5. Worked Examples
Example: (0,0,0),(1,0,0),(0,1,0): cross=(0,0,1)≠0, unique plane z=0.
6. Common Pitfalls
Use the cross product, not dot;
Keep points distinct;
Collinear points admit infinitely many planes.

❓ FAQ | Points, Planes

Always a plane?
Only if non-collinear.
Collinearity?
|AB×AC|=0.
Corollaries?
Line + point.
Relations?
Contain, intersect, parallel.
Two planes?
A line.
Cross magnitude?
Parallelogram area.
Collinear?
No unique plane.