Calculator
Example Data Table
| P1 | P2 | P3 | General form | Normal vector |
|---|---|---|---|---|
| (1, 2, 3) | (4, 0, 1) | (-2, 5, 2) | 8x + 9y + 3z - 35 = 0 | <8, 9, 3> |
| (0, 0, 0) | (1, 0, 0) | (0, 1, 0) | z = 0 | <0, 0, 1> |
| (2, -1, 4) | (5, 3, 0) | (1, 2, 6) | 20x - 10y + 13z - 102 = 0 | <20, -10, 13> |
Formula Used
Step 1: Use the points P1(x1, y1, z1), P2(x2, y2, z2), and P3(x3, y3, z3).
Step 2: Build vectors u = P2 - P1 and v = P3 - P1.
Step 3: Find the normal vector with n = u × v = <A, B, C>.
Step 4: Put P1 into A(x - x1) + B(y - y1) + C(z - z1) = 0.
Final form: Ax + By + Cz + D = 0, where D = -(Ax1 + By1 + Cz1).
Distance from test point Q: |Aqx + Bqy + Cqz + D| / sqrt(A² + B² + C²).
How to Use This Calculator
- Enter the x, y, and z coordinates for all three points.
- Add a test point if you want distance and projection details.
- Select the decimal precision for rounded results.
- Press the calculate button to get the general equation.
- Review the vectors, normal vector, intercepts, and graph.
- Use CSV or PDF buttons to save your result.
What This Calculator Does
A plane in three dimensional geometry is fixed by three non collinear points. This calculator finds that plane and shows each important form. It first builds two direction vectors from the given points. Then it takes their cross product. That cross product becomes the normal vector. The normal vector controls the final equation.
Why Three Points Matter
One point is not enough. Two points define a line. Three valid points define a flat surface. The points must not sit on the same straight line. When they are collinear, the cross product becomes zero. Then infinitely many planes can pass through the same line. The tool checks this condition before showing results.
Useful Output Forms
The standard plane form is Ax + By + Cz + D = 0. The point normal form is A(x − x1) + B(y − y1) + C(z − z1) = 0. The determinant form is also useful for proofs. The calculator also gives a normalized equation. It divides coefficients by the normal length. That makes distance checks easier.
Graph and Visual Check
The graph helps confirm the result. It plots the three points. It also draws a small patch of the plane. This visual step is helpful when signs look confusing. You can rotate the graph and compare the positions of the points. All three points should appear on the same surface.
Advanced Geometry Uses
Extra fields let you test another point. The tool calculates signed distance, absolute distance, and the projection of that point on the plane. These values are useful in vectors, analytic geometry, physics, graphics, and engineering. Triangle area is also displayed because it comes from half the normal vector length.
Exporting Your Work
Use the CSV button to save numerical results. Use the PDF button to create a clean report. Both exports include the main equation, vector details, and optional point analysis. Keep the precision setting high for exact checking. Lower precision is better for short homework answers.
Checking Accuracy
Substitute each original point into the final equation. The left side should become zero, except for rounding. This simple check catches typing mistakes, swapped coordinates, and wrong signs before you submit work with confidence.
FAQs
Can three points always define a plane?
No. The three points must be different and non collinear. If they lie on one straight line, many planes can pass through that line, so one unique plane is not defined.
What is the general equation of a plane?
The general equation is Ax + By + Cz + D = 0. The values A, B, and C form a normal vector. D shifts the plane through the selected point.
Why is the cross product used?
The cross product creates a vector perpendicular to two direction vectors on the plane. That perpendicular vector is the normal vector used in the plane equation.
What does a zero normal vector mean?
A zero normal vector means the direction vectors are parallel or one vector has no length. This usually means repeated points or collinear points were entered.
How is distance from a point calculated?
The calculator substitutes the test point into Ax + By + Cz + D, then divides by the normal vector length. The absolute value gives the shortest distance.
What is the normalized plane equation?
It is the same plane equation after dividing all coefficients by the normal length. This makes the normal vector have a length of one.
Can I use decimal coordinates?
Yes. The fields accept integers, decimals, and negative values. You can change decimal precision to control how many digits appear in the final answer.
Why should I use the graph?
The graph gives a quick visual check. It shows whether the three points sit on the same displayed plane and helps detect input mistakes.