Enter two three-dimensional vectors
Choose the order, set precision, then calculate the normal vector and included angle.
Worked vector example
| Vector A | Vector B | A × B | Magnitude | Angle | Use case |
|---|---|---|---|---|---|
| ⟨1, 2, 3⟩ | ⟨4, 5, 6⟩ | ⟨−3, 6, −3⟩ | 7.348469 | 12.933° | General vector comparison |
| ⟨1, 0, 0⟩ | ⟨0, 1, 0⟩ | ⟨0, 0, 1⟩ | 1 | 90° | Perpendicular axes |
| ⟨2, 4, 6⟩ | ⟨1, 2, 3⟩ | ⟨0, 0, 0⟩ | 0 | 0° | Parallel direction check |
Core vector relationships
A × B = ⟨AᵧB_z − A_zBᵧ, A_zBₓ − AₓB_z, AₓBᵧ − AᵧBₓ⟩
|A × B| = |A||B| sin θ
θ = cos⁻¹((A · B) / (|A||B|))
The dot product is A · B = AₓBₓ + AᵧBᵧ + A_zB_z. The cross-product magnitude gives parallelogram area. Half of that magnitude gives triangle area.
Get a reliable vector result
- Enter the x, y, and z components of vector A.
- Enter the matching components of vector B.
- Choose A × B or B × A. The magnitude remains the same.
- Select the number of decimal places you need.
- Press the calculate button. Review the vector, magnitude, angle, and relation.
- Use CSV or PDF export when you need a saved record.
Understanding Cross Products and Vector Angles
Cross products describe a vector created from two three-dimensional vectors. The new vector is perpendicular to both original vectors. Its direction follows the right-hand rule. Its magnitude measures how strongly the vectors spread apart. This makes the operation useful in geometry, physics, engineering, graphics, and navigation.
The calculator accepts the x, y, and z components for vectors A and B. It finds A × B by evaluating a determinant. The first component uses AyBz minus AzBy. The second uses AzBx minus AxBz. The third uses AxBy minus AyBx. Those three values form the normal vector.
Magnitude matters because it represents parallelogram area. The magnitude of A × B equals |A||B|sin θ. Here, θ is the smaller angle between the two vectors. Divide the cross-product magnitude by two for triangle area. A zero magnitude means the vectors are parallel, anti-parallel, or one vector has zero length.
Angle calculations use the dot product. First find A · B. Then divide that result by |A||B|. The inverse cosine of the quotient gives the angle. The calculator reports degrees and radians. It also shows whether the pair is perpendicular, parallel, anti-parallel, or generally positioned. A zero-length vector has no defined direction or angle.
Use consistent units for both vectors. Components may represent metres, forces, velocities, or any compatible quantity. The cross product then receives combined units. For example, position cross force produces torque. The result direction identifies the torque axis. A positive or negative component alone does not decide the orientation. Read all components together.
The plot shows vectors A and B from the origin. It also displays a scaled cross-product arrow. Scaling keeps the normal arrow visible when its raw magnitude is very large. The chart is a visual aid. Use the listed numeric values for exact work. The component table and formula steps help you check each stage.
Export tools make repeated work easier. CSV creates a compact data file for spreadsheets. PDF creates a clean result summary for reports or homework. Keep enough decimal places for your task. Round only after reviewing the full value. Enter known values, calculate, inspect the angle, and save the results when needed.
Use the unit normal when a direction alone is required. It has length one and preserves the cross-product orientation for coordinate transformations directly.
Cross product calculator FAQs
1. What does a cross product calculate?
It calculates a vector perpendicular to two three-dimensional input vectors. Its magnitude also equals the area of the parallelogram formed by those vectors.
2. Why does order matter?
A × B and B × A have identical magnitudes. Their directions are opposite. Switching the order changes every cross-product component to its negative.
3. Can this calculator find an angle in degrees?
Yes. It calculates the included angle with the dot-product formula and reports both degrees and radians whenever both vectors have nonzero length.
4. What happens when vectors are parallel?
Their cross product is the zero vector. The magnitude is zero because parallel vectors do not create a parallelogram with positive area.
5. What is the right-hand rule?
Point your right-hand fingers along the first vector, then curl them toward the second. Your thumb indicates the direction of the first cross the second.
6. Can I use negative vector components?
Yes. Negative components are valid. They indicate direction along a negative coordinate axis and are handled normally by the component formulas.
7. Why is the angle undefined for a zero vector?
A zero vector has no direction. The angle formula divides by both vector magnitudes, so an angle cannot be determined when either magnitude is zero.
8. What does the unit normal vector show?
It shows only the normal direction. Its magnitude is one. It is useful when orientation matters but the original cross-product size does not.
9. Are cross products defined in two dimensions?
The standard cross product uses three-dimensional vectors. A two-dimensional version is often treated as the z component of a three-dimensional result.
10. Which units should I enter?
Use consistent component units. For example, use metres for both position vectors. The cross product then uses squared or combined units, depending on the quantities.
11. Can I export the calculation?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a concise report-ready summary.