What is the Cross Product?
Cross Product Calculator: The cross product (also known as the vector product) of two vectors in three-dimensional space results in another vector that is perpendicular to both of the original vectors. This operation is used extensively in physics and engineering, particularly in determining torque and rotational force. The magnitude of the cross product vector is equal to the area of the parallelogram formed by the two original vectors.
Vector A
| x | y | z |
|---|---|---|
Vector B
| x | y | z |
|---|---|---|
Result
| x | y | z |
|---|---|---|
| 0 | 0 | 0 |
Method of Solution
Please calculate using the above inputs.
Result Chart
How to Use the Cross Product Calculator
To use the Cross Product Calculator, input the x, y, and z components of two vectors into the respective fields. Click the "Calculate" button to compute the cross product. The result will display below the form. To reset the fields and results, click the "Clear" button. This tool simplifies finding the vector perpendicular to two given vectors.
Formula of the Cross Product
The formula for the cross product of two vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz) is given by:
A × B = (Ay * Bz - Az * By, Az * Bx - Ax * Bz, Ax * By - Ay * Bx)
1. What does the cross product represent?
The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the original vectors. It has applications in calculating torque, rotational forces, and in determining the orientation of vectors in space.
2. Can the cross product be zero?
Yes, the cross product can be zero if the two vectors are parallel or if one of the vectors is a zero vector. In such cases, there is no unique direction perpendicular to both vectors, leading to a zero vector result.
3. Is the cross product commutative?
No, the cross product is not commutative. Swapping the order of the vectors results in a vector pointing in the opposite direction. Specifically, A × B = - (B × A).
4. What is the magnitude of the cross product vector?
The magnitude of the cross product vector represents the area of the parallelogram formed by the two original vectors. It is calculated as |A × B| = |A| |B| sin(θ), where θ is the angle between the vectors.
5. Can the cross product be used in 2D space?
While the cross product is typically defined in 3D space, a similar concept can be applied in 2D by extending vectors into 3D. In 2D, the cross product's magnitude corresponds to the signed area of the parallelogram formed by the vectors.
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