What is a Cross Product Calculator?
The Cross Product Calculator computes the cross product of two 3D vectors, yielding a new vector that is perpendicular to both input vectors. This is useful in physics and engineering to determine forces or directions orthogonal to a plane formed by the original vectors.
Formula
The cross product of vectors A and B is given by:
C = A × B = (a_y * b_z - a_z * b_y, a_z * b_x - a_x * b_z, a_x * b_y - a_y * b_x)
How to Use This Calculator
Enter the x, y, and z components of Vector A and Vector B in the provided fields. Click "Calculate" to get the cross product vector. The result will be displayed along with the calculation details. Use the "Clear" button to reset the fields and start over.
Vector A
| x | y | z |
|---|---|---|
Vector B
| x | y | z |
|---|---|---|
Frequently Asked Questions
1. What is the cross product used for?
The cross product is used in physics and engineering to find a vector perpendicular to two given vectors. It is useful for calculating torque, angular momentum, and determining the orientation of a plane.
2. How is the cross product different from the dot product?
The dot product results in a scalar value representing the magnitude of the projection of one vector onto another. In contrast, the cross product results in a vector perpendicular to both input vectors, with magnitude related to the area of the parallelogram formed by them.
3. Can the cross product be computed for 2D vectors?
While the cross product is primarily defined for 3D vectors, in 2D vectors, it can be extended by considering them in 3D space with a z-component of zero. The result will be a scalar representing the magnitude of the perpendicular vector in 3D space.
4. What is the geometric interpretation of the cross product?
The cross product of two vectors represents a vector that is perpendicular to the plane formed by the two vectors. Its magnitude equals the area of the parallelogram formed by the vectors, and its direction is given by the right-hand rule.
5. How do I compute the magnitude of the cross product?
The magnitude of the cross product can be computed using the formula: |C| = |A| * |B| * sin(θ), where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them. The magnitude represents the area of the parallelogram spanned by A and B.
6. Is the cross product commutative?
No, the cross product is not commutative. This means that A × B is not equal to B × A. Instead, A × B = - (B × A), which indicates that the cross product changes direction if the order of vectors is swapped.
7. What happens if the two vectors are parallel?
If two vectors are parallel, their cross product is zero because the sine of the angle between them is zero. This means no vector is perpendicular to both, as they are in the same or opposite direction.
8. How can I visualize the cross product result?
The result of the cross product can be visualized as a vector that is orthogonal to the plane containing the original vectors. In 3D space, it can be represented graphically using 3D plotting tools or software.
9. Can the cross product be used in 4D space?
The concept of the cross product is specific to 3D space. In higher dimensions, there are other methods like the wedge product or exterior algebra to compute analogous quantities, but they do not directly generalize the cross product.
10. How do I interpret the direction of the cross product vector?
The direction of the cross product vector is determined by the right-hand rule: point your right-hand fingers in the direction of the first vector and curl them towards the second vector. Your thumb points in the direction of the cross product vector.
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