Cross Product Calculator
Calculate the cross product of two 3D vectors. Find the resultant vector, magnitude, angle between vectors, and parallelogram area, step by step.
Quick Examples
Vector A
Vector B
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Cross Product Calculator
Calculate the cross product of two 3D vectors instantly. Our calculator shows the resultant vector in component form and i,j,k notation, computes magnitude, angle between vectors, and the area of the parallelogram formed by the vectors. Includes step-by-step solutions using the determinant method.
What is a Cross Product?
The cross product (vector product) of two vectors a and b produces a third vector c that is perpendicular to both input vectors. Unlike the dot product which gives a scalar, the cross product gives a vector. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, calculated as |a||b|sin(θ) where θ is the angle between them. The direction of the result follows the right-hand rule.
Cross Product Formula
a × b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)kHow to Calculate Cross Product
Enter the x, y, and z components for Vector A (Aₓ, Aᵧ, Aᵤ)
Enter the x, y, and z components for Vector B (Bₓ, Bᵧ, Bᵤ)
View the cross product result in both component and i,j,k notation
Check the magnitude and angle between your vectors
Use the swap button to see a × b vs b × a (demonstrates anti-commutative property)
Cross Product Applications
Torque Calculation
Calculate rotational force as τ = r × F, where r is the position vector and F is the applied force.
Magnetic Force
Find the force on a charged particle as F = qv × B in electromagnetic physics problems.
Surface Normal Vectors
Compute perpendicular vectors to surfaces for 3D graphics, lighting, and rendering.
Area of Parallelogram
The magnitude |a × b| gives the area of the parallelogram formed by vectors a and b.
Angular Momentum
Calculate angular momentum L = r × p in classical mechanics problems.
Plane Equations
Find the normal vector to a plane defined by two direction vectors.
Why Use This Calculator
Avoid Calculation Errors
Cross product calculations involve multiple terms with negative signs. Our calculator ensures accuracy every time.
Physics Applications
Essential for calculating torque (τ = r × F), magnetic force (F = qv × B), and angular momentum.
Computer Graphics
Compute surface normal vectors for 3D rendering, lighting calculations, and collision detection.
Engineering Problems
Solve mechanics problems involving rotational forces, moments, and perpendicular vector calculations.
Step-by-Step Learning
See the complete determinant expansion and understand how each component is calculated.
Frequently Asked Questions
The dot product (a · b) produces a scalar (single number) and measures how much two vectors point in the same direction. The cross product (a × b) produces a vector that is perpendicular to both inputs. Dot product uses cosine of the angle; cross product uses sine.
Point your fingers in the direction of vector a, curl them toward vector b, and your thumb points in the direction of a × b. For i × j = k: point fingers along x-axis, curl toward y-axis, thumb points along z-axis.
Yes, treat 2D vectors as 3D with z = 0. The cross product of two 2D vectors gives a vector along the z-axis only, essentially a scalar representing the signed area. Use our 2D mode for this calculation.
No, cross product is anti-commutative: a × b = -(b × a). Swapping the order reverses the direction of the result. This is why order matters when calculating torque or magnetic force.
Set up a 3×3 determinant with i, j, k in the first row, components of a in the second row, and components of b in the third row. Expand along the first row to get: i(a₂b₃-a₃b₂) - j(a₁b₃-a₃b₁) + k(a₁b₂-a₂b₁).
A zero cross product means the vectors are parallel (or anti-parallel). Since the magnitude equals |a||b|sin(θ), when θ = 0° or 180°, sin(θ) = 0, making the cross product zero.