Cross Product Calculator: Visualize 3D Vector Products
Calculate the cross product of two 3D vectors with our interactive online tool. Visualize vectors and results in a dynamic 3D plot. Perfect for students, engineers, and researchers.
Enter Your Vectors
Define vectors A and B by entering their x, y, and z components.
Vector A
Vector B
Cross Product Result
Formula:
Result Vector:
Understanding Cross Product
The cross product, also known as the vector product, is a fundamental operation in vector algebra, specifically in three-dimensional space. Given two vectors, it produces a third vector that is perpendicular to both of them.
Magnitude: The length of the cross product vector is equal to the area of the parallelogram formed by the two original vectors. It's calculated as \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin(\theta) \), where \( \theta \) is the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\).
Direction: The direction of the cross product is perpendicular to both input vectors, determined by the right-hand rule. If you curl the fingers of your right hand from vector \(\mathbf{a}\) to vector \(\mathbf{b}\), your thumb points in the direction of \(\mathbf{a} \times \mathbf{b}\).
Applications: Cross product is crucial in physics and engineering for:
- Torque Calculation: In mechanics, torque is calculated using the cross product of force and lever arm.
- Magnetic Force: In electromagnetism, the force on a moving charge in a magnetic field is given by a cross product.
- Surface Normals: In computer graphics and 3D geometry, cross products are used to find surface normals for lighting and shading.
For further reading, visit Wikipedia on Cross Product.