Calculator Inputs
Enter the components for two three-dimensional vectors. The form uses a single-column page layout, while inputs respond across screen sizes.
Plotly Graph
The chart compares vector A, vector B, and the resulting cross product by component.
Formula Used
For a vector A = (ax, ay, az), the cross product matrix, also called the skew-symmetric matrix, is:
[A]× = [ [0, -az, ay], [az, 0, -ax], [-ay, ax, 0] ]
For vector B = (bx, by, bz), the matrix identity is:
A × B = [A]×B
The expanded component formula is:
A × B = (aybz - azby, azbx - axbz, axby - aybx)
The magnitudes and angle use:
|A| = √(ax2 + ay2 + az2)|B| = √(bx2 + by2 + bz2)θ = cos-1((A·B)/(|A||B|))
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.
- Choose the number of decimal places for displayed answers.
- Click Calculate Now to generate the matrix and vector results.
- Review the result panel placed below the header and above the form.
- Use the chart to compare the component sizes visually.
- Download the summary as CSV or PDF when needed.
Example Data Table
This worked example uses the page defaults.
| Vector A | Vector B | Skew Matrix of A | A × B | Dot Product | Angle (deg) |
|---|---|---|---|---|---|
| (3, -2, 5) | (4, 1, -1) | [[0, -5, -2], [5, 0, -3], [2, 3, 0]] | (-3, 23, 11) | 5 | 78.9784 |
Frequently Asked Questions
1. What is a cross product matrix?
A cross product matrix is a skew-symmetric matrix built from one vector. Multiplying that matrix by another vector produces the same result as the vector cross product in three dimensions.
2. Why is the matrix skew-symmetric?
Its transpose equals its negative. That structure preserves the algebra needed to represent rotational behavior and cross products using matrix multiplication instead of only determinant-style expansion.
3. When does the cross product become zero?
The cross product becomes zero when the vectors are parallel, anti-parallel, or when at least one vector is the zero vector. In those cases, no unique perpendicular direction exists.
4. Can this calculator verify A × B using matrix multiplication?
Yes. It computes the skew-symmetric matrix of vector A, multiplies it by vector B, and shows that result beside the direct cross product for confirmation.
5. Why does the skew matrix determinant show zero?
Every 3×3 skew-symmetric matrix has determinant zero. That means it is singular and cannot be inverted in the ordinary matrix sense.
6. What does the angle result mean?
The angle is the geometric separation between the two vectors. It is found from the dot product and magnitudes, then reported here in degrees for easier interpretation.
7. Is this calculator only for three-dimensional vectors?
Yes. The standard vector cross product and its skew-symmetric matrix form are defined here for three-dimensional vectors only.
8. Where is this method used in mathematics and engineering?
It appears in linear algebra, rigid body motion, robotics, mechanics, graphics, and coordinate transformations. The matrix form is especially useful in compact derivations and numerical implementations.