Enter Vector Components
The page uses one main content column. The calculator inputs shift across 3, 2, and 1 columns by screen size.
Example Data Table
This sample demonstrates how order changes the triple product result.
| Vector A | Vector B | Vector C | A × (B × C) | (A × B) × C |
|---|---|---|---|---|
| (1, 2, 3) | (4, -1, 2) | (0, 5, -2) | (16, -44, 24) | (25, 14, 35) |
Formula Used
The vector triple product follows the BAC–CAB identities below.
A × (B × C) = B(A · C) − C(A · B)
(A × B) × C = B(A · C) − A(B · C)
The calculator also computes the direct cross products first, then compares those answers with the identity-based expansions. A tolerance value confirms whether both methods agree after rounding effects.
How to Use This Calculator
- Enter the x, y, and z components of vectors A, B, and C.
- Select whether to compute the left product, right product, or both.
- Choose the number of decimal places for displayed values.
- Set a tolerance for checking identity verification.
- Press the calculate button to show results above the form.
- Use the export buttons to save the current output as CSV or PDF.
FAQs
1. What does this calculator solve?
It computes vector triple products from three input vectors. You can evaluate A × (B × C), (A × B) × C, or both together, while viewing intermediate steps and verification details.
2. Why are the two triple products different?
Cross products are not associative. That means changing the grouping changes the result. This calculator shows both products so you can compare direction, magnitude, and identity expansion side by side.
3. What is the BAC–CAB identity?
It is a standard vector identity that rewrites a triple cross product using dot products and scaled vectors. It helps simplify derivations and provides a reliable way to verify numerical calculations.
4. Why is a tolerance setting included?
Tolerance helps compare direct and identity-based results when decimal rounding is present. Very small floating-point differences can appear, so verification should allow a tiny numerical margin.
5. What happens if the result is the zero vector?
The magnitude becomes zero, and the unit vector is undefined. The calculator states this clearly instead of forcing a division by zero during normalization.
6. Can I use decimal or negative inputs?
Yes. Each component field accepts positive values, negative values, and decimals. This supports classroom problems, analytic geometry, engineering models, and physics applications.
7. What do the export buttons save?
The CSV and PDF downloads include the current vectors, dot products, intermediate cross products, both triple product results, verification status, and comparison values from the latest calculation.
8. Where is this calculator most useful?
It is useful in vector algebra, mechanics, electromagnetics, geometry, and computational modelling. Teachers, students, and analysts can use it to test identities and document worked examples quickly.