Compute scalar and vector triple products from components. See determinants, identities, orientation, and magnitude details. Great for mechanics, electromagnetism, coordinate systems, and vector analysis.
Enter three vectors, choose the operation type, set formatting, and calculate scalar or vector triple products.
The scalar triple product measures signed volume and orientation.
A · (B × C) A · (B × C) = det [ A ; B ; C ]Its absolute value gives the volume of the parallelepiped formed by A, B, and C. A zero value means the vectors are coplanar.
The vector triple product returns another vector and follows the BAC minus CAB identity.
A × (B × C) A × (B × C) = B(A · C) − C(A · B)This identity avoids computing two cross products directly and is useful in many physics derivations involving forces, fields, and rotational systems.
These sample entries show how scalar and vector triple products behave for different vector sets.
| Vector A | Vector B | Vector C | A · (B × C) | A × (B × C) | Interpretation |
|---|---|---|---|---|---|
| (1, 2, 3) | (4, 0, 1) | (2, -1, 5) | -47 | (46, 7, -20) | Nonzero volume with negative orientation. |
| (2, 1, 0) | (1, 0, 1) | (3, 1, 1) | 0 | (1, -2, 5) | Coplanar vectors because scalar result is zero. |
| (0, 1, 2) | (1, 2, 0) | (2, 1, 1) | -7 | (-1, 4, -2) | Useful for checking handedness and identity form. |
It represents the signed volume of the parallelepiped formed by three vectors. Its sign also tells you the orientation of the ordered vector set.
It becomes zero when the three vectors are coplanar or linearly dependent. In that case, they do not span a three-dimensional volume.
A negative value means the ordered set of vectors has opposite orientation from a right-handed system. The magnitude still measures volume.
It is A × (B × C) = B(A · C) − C(A · B). This identity simplifies derivations and often avoids repeated cross-product expansion.
Comparing both forms verifies the calculation. If they match, the result is consistent with the standard vector triple product identity.
If all input vectors share the same unit, the displayed triple-product results are shown with a cubic-style unit label for convenience and readability.
Yes. The calculator accepts integers, decimals, and negative values, which is helpful for real physics problems and coordinate-system analysis.
The graph shows vector direction and relative size in 3D space. It helps you visually inspect the entered vectors and computed vector results.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.