Cross Product Calculator – Guide & Formulas
Calculate the cross product of two 3D vectors instantly. Enter vector components to see the result vector, magnitude, area of parallelogram, and step-by-step solution.
Calculate the cross product of two 3D vectors with our free online calculator. Enter components to see the result vector, magnitude, area, and step-by-step solution.
Key Takeaway
Use the free Cross Product Calculator to calculate the cross product of two 3d vectors instantly. enter vector components to see the result vector, magnitude, area of parallelogram, and step-by-step solution. Get instant results with step-by-step explanations.
How to Use the Cross Product Calculator
- Step 1: Enter the components of the first vector (x, y, z)
- Step 2: Enter the components of the second vector (x, y, z)
- Step 3: Click Calculate to see the cross product vector
- Step 4: Review the determinant expansion, magnitude, and area of the parallelogram
The Formula
Variable Definitions
- A × B: The cross product of vectors A and B — produces a new vector
- î, ĵ, k̂: Unit vectors along the x, y, and z axes
- Ax, Ay, Az: Components of vector A along x, y, z axes
- Bx, By, Bz: Components of vector B along x, y, z axes
- |A × B|: Magnitude of cross product — equals the area of the parallelogram formed by A and B
Cross Product of A = [1, 2, 3] and B = [4, 5, 6]
Compute A × B using the determinant method.
- Step 1: Set up the determinant: |î ĵ k̂; 1 2 3; 4 5 6|.
- Step 2: î component: (2×6 - 3×5) = 12 - 15 = -3.
- Step 3: ĵ component: -(1×6 - 3×4) = -(6 - 12) = 6.
- Step 4: k̂ component: (1×5 - 2×4) = 5 - 8 = -3.
- Step 5: A × B = [-3, 6, -3]. Magnitude = √(9+36+9) = √54 ≈ 7.348.
Frequently Asked Questions
What is the cross product?
The cross product of two vectors A and B produces a new vector that is perpendicular to both A and B. Its magnitude equals the area of the parallelogram formed by the two vectors.
When is the cross product zero?
The cross product is the zero vector when A and B are parallel (angle = 0° or 180°) or when either vector is the zero vector. Parallel vectors have no perpendicular direction.
What does the cross product magnitude represent?
The magnitude |A × B| equals the area of the parallelogram with sides A and B. It also equals |A| × |B| × sin(θ), where θ is the angle between them.
Why is the cross product only for 3D?
The cross product is defined in 3D because it requires a third dimension for the result to be perpendicular to both input vectors. In 2D, the result is a scalar (pseudo-scalar).
What is the right-hand rule?
The right-hand rule determines the direction of the cross product. Point your fingers along A, curl them toward B, and your thumb points in the direction of A × B.
How is the cross product used in physics?
Cross products appear in torque (τ = r × F), angular momentum (L = r × p), magnetic force (F = qv × B), and many other physics applications involving rotation and perpendicular forces.