Scalar Triple Product Calculator
Enter three 3D vectors. The tool returns the determinant, orientation, coplanarity result, and volume measures.
Plotly Graph
The graph compares x, y, and z components of the three vectors.
Example Data Table
| Vector A | Vector B | Vector C | Scalar Triple Product | Absolute Volume | Coplanar |
|---|---|---|---|---|---|
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 | 1 | No |
| (2, 1, 0) | (1, 3, 2) | (4, 0, 1) | 13 | 13 | No |
| (3, -1, 2) | (0, 4, 1) | (2, 1, 5) | 39 | 39 | No |
| (1, 2, 3) | (2, 4, 6) | (0, 1, 1) | 0 | 0 | Yes |
Formula Used
The scalar triple product measures the signed volume generated by three vectors. It is found by taking the dot product of one vector with the cross product of the other two.
Geometric meaning: The absolute value equals the parallelepiped volume.
Zero result: A zero value means the vectors are coplanar.
Sign: The sign shows whether the ordered vectors follow a positive or negative orientation.
How to Use This Calculator
- Enter the x, y, and z components for vector A.
- Enter the x, y, and z components for vector B.
- Enter the x, y, and z components for vector C.
- Click Calculate Now to compute the determinant-based result.
- Read the signed product, absolute volume, and tetrahedron volume.
- Check the coplanarity status and orientation label.
- Review the plotted component comparison for all vectors.
- Use the CSV or PDF buttons to save the result.
FAQs
1. What does the scalar triple product represent?
It represents the signed volume formed by three vectors in 3D space. Its absolute value gives the parallelepiped volume, while the sign shows orientation.
2. When does the result become zero?
The result becomes zero when the three vectors lie in the same plane. That means they are coplanar and do not create a three-dimensional volume.
3. Why can the scalar triple product be negative?
A negative value appears when the ordered vectors follow a negative orientation. The magnitude still shows volume, but the sign records vector order.
4. Does changing vector order affect the answer?
Yes. Reordering vectors can change the sign. Cyclic changes keep the same value, while swapping two vectors reverses the sign.
5. Can I use decimal values?
Yes. The calculator accepts integers and decimals. This helps when vectors come from measurements, simulations, or coordinate geometry problems.
6. What is the difference between signed and absolute volume?
Signed volume keeps the positive or negative orientation. Absolute volume removes the sign and reports only the physical size of the 3D shape.
7. How is tetrahedron volume related here?
A tetrahedron built from the same three vectors has volume equal to one-sixth of the absolute scalar triple product. This tool shows that extra measure.
8. How can I verify the result manually?
Compute B × C first. Then take the dot product with A. You can also evaluate the 3×3 determinant formed by the three vector rows.