Enter Vector Values
Example Data Table
| Vector A | Vector B | A × B | |A × B| | Angle | Area Meaning |
|---|---|---|---|---|---|
| <3, -3, 1> | <4, 9, 2> | <-15, -2, 39> | 41.8330 | 111.4376° | Parallelogram area |
| <1, 0, 0> | <0, 1, 0> | <0, 0, 1> | 1.0000 | 90.0000° | Unit square area |
| <2, 2, 0> | <4, 4, 0> | <0, 0, 0> | 0.0000 | 0.0000° | Parallel vectors |
Formula Used
B = <Bx, By, Bz>
A × B = <(Ay × Bz - Az × By), (Az × Bx - Ax × Bz), (Ax × By - Ay × Bx)>
|A × B| = sqrt(Cx² + Cy² + Cz²)
Area of parallelogram = |A × B|
Area of triangle = |A × B| / 2
θ = cos⁻¹((A · B) / (|A||B|))
The cross product creates a new vector perpendicular to both input vectors. Its direction follows the right hand rule. Its magnitude equals the area of the parallelogram formed by the two vectors.
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.
- Add Vector C if you want scalar triple product analysis.
- Use scale values when vectors need multiplication first.
- Select the decimal precision for cleaner output.
- Press the calculate button to view results above the form.
- Download the result as a CSV file or PDF report.
About Cross Product Calculations
Purpose Of The Calculator
A cross product calculator helps solve three dimensional vector problems with speed and clarity. It is useful in geometry, mechanics, graphics, robotics, mapping, and many engineering tasks. The tool accepts two vectors and returns a perpendicular vector. This result is called the normal vector. The normal vector can describe surface direction, rotation axis, or orientation.
Why Direction Matters
The cross product is not only a number. It is a vector with direction and length. The direction follows the right hand rule. Curl your fingers from Vector A toward Vector B. Your thumb shows the direction of A cross B. Reversing the order changes the sign. That is why A cross B and B cross A point in opposite directions.
Area And Magnitude
The magnitude of the cross product has a direct geometric meaning. It equals the area of the parallelogram built from both vectors. Half of that value gives the triangle area. This makes the calculator helpful for coordinate geometry. It also helps when finding torque, moment, and surface normal strength.
Advanced Vector Review
This calculator also shows the dot product and angle between vectors. These values help compare direction and alignment. A small angle means the vectors point nearly together. A right angle means they are perpendicular. The optional third vector gives scalar triple product output. That value can represent signed volume in vector geometry.
Practical Use
Use this tool when manual component expansion feels slow or error prone. Enter clean components, check the normal vector, and review the area values. The export buttons help save results for homework, reports, lab notes, or project records. The calculator keeps every result organized and easy to compare.
FAQs
1. What is a cross product?
A cross product is a vector operation for three dimensional vectors. It returns a vector perpendicular to both input vectors.
2. Why does order matter in cross product?
Order matters because A × B points opposite to B × A. The magnitude remains the same, but direction changes.
3. What does the magnitude mean?
The magnitude equals the area of the parallelogram formed by the two vectors. Half of it gives triangle area.
4. Can the cross product be zero?
Yes. It becomes zero when one vector is zero or both vectors are parallel. Then no parallelogram area is formed.
5. What is a unit normal vector?
A unit normal vector is the cross product direction scaled to length one. It shows direction without size influence.
6. What is the scalar triple product?
The scalar triple product is A · (B × C). It can represent signed volume made by three vectors.
7. Does this calculator handle decimal values?
Yes. You can enter integers, decimals, negative values, and scaled vectors. You can also set decimal precision.
8. What can I export?
You can export the main vector results, magnitudes, angle, area values, dot product, and triple product summary.