Cross Product Calculator
Compute the cross product of two vectors, find the magnitude, direction, and angle between vectors. Enter your vector components and get step-by-step solutions using the determinant method.
Vector A
Vector B
Understanding the Cross Product
The cross product is a fundamental operation in vector algebra that takes two vectors in three-dimensional space and produces a new vector perpendicular to both. Unlike the dot product which yields a scalar, the cross product gives a vector result, making it essential for computations involving torque, angular momentum, and surface normals.
Cross Product Formula
A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
The cross product is computed component-by-component using the differences of products of the input vector entries. This formula is equivalent to expanding the determinant of a 3×3 matrix with unit vectors in the first row.
Key Properties
A × B = -(B × A)
The cross product is anti-commutative: swapping the operands reverses the direction. It is also distributive over addition and compatible with scalar multiplication, but it is not associative.
Right-Hand Rule
Direction: fingers A → curl to B → thumb = A × B
The direction of A × B follows the right-hand rule. Point your index finger along A, curl your fingers toward B, and your thumb points in the direction of the cross product. This convention is standard in physics and engineering.
Applications
|A × B| = |A||B|sin(θ)
The magnitude of the cross product equals the area of the parallelogram formed by A and B. This property is used for computing areas, torque (τ = r × F), magnetic force (F = qv × B), and surface normals in 3D graphics.
Cross Product Calculation Methods
There are several equivalent ways to compute the cross product. Each method gives the same result but offers different insights into the operation. Our calculator uses the component formula internally and displays the determinant expansion for educational clarity.
3D Component Formula
i: a₂b₃ - a₃b₂
j: a₃b₁ - a₁b₃
k: a₁b₂ - a₂b₁
The most direct method. Compute each component of the result vector separately using pairs of components from the input vectors. Each component is a 2×2 determinant formed by deleting the corresponding row and column.
Component Form
A = a₁i + a₂j + a₃k
B = b₁i + b₂j + b₃k
Express each vector in terms of unit vectors i, j, k, then distribute the cross product using i×j = k, j×k = i, k×i = j, and the anti-commutativity property. Collect like terms to get the final result. This approach helps build intuition about the operation.
Determinant Method
|i j k |
|a₁ a₂ a₃|
|b₁ b₂ b₃|
Write a 3×3 matrix with unit vectors in the top row, then expand along the first row using cofactor expansion. This gives the same component formula but is easier to memorize and less error-prone. It is the most popular textbook method.
Cross Product Applications
The cross product is one of the most widely used vector operations across science and technology. Its ability to produce a perpendicular vector and encode area information makes it indispensable in many fields.
Physics
- • Torque calculation: τ = r × F
- • Angular momentum: L = r × p
- • Magnetic force on a moving charge: F = qv × B
- • Electromagnetic field interactions (Poynting vector)
Engineering
- • Structural analysis of forces and moments
- • Robotics: joint axis and motion planning
- • Fluid dynamics: vorticity and circulation
- • Mechanical design and torque specifications
Computer Graphics
- • Surface normal computation for lighting and shading
- • Back-face culling in 3D rendering pipelines
- • Collision detection between 3D objects
- • Camera orientation and view transformations
Geometry
- • Area of parallelogram: |A × B|
- • Area of triangle: ½|A × B|
- • Volume of parallelepiped: |A · (B × C)|
- • Testing collinearity and coplanarity of points
Frequently Asked Questions
What is the cross product?
The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space. Given two vectors A and B, their cross product A × B produces a third vector that is perpendicular to both A and B. The magnitude of the resulting vector equals |A||B|sin(θ), where θ is the angle between A and B. The cross product is fundamental in physics, engineering, and computer graphics for computing torque, angular momentum, surface normals, and areas.
How do you calculate a cross product?
To calculate the cross product of vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), use the formula: A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁). This can be remembered using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row, vector A components in the second row, and vector B components in the third row. Our calculator performs this computation instantly and shows each step.
What is the formula for cross product of two vectors?
The cross product formula for A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) is: A × B = i(a₂b₃ - a₃b₂) - j(a₁b₃ - a₃b₁) + k(a₁b₂ - a₂b₁). The magnitude is |A × B| = |A||B|sin(θ). This formula comes from expanding the determinant |i j k; a₁ a₂ a₃; b₁ b₂ b₃| along the first row. Each component uses a 2×2 minor determinant.
What is the difference between dot product and cross product?
The dot product and cross product are two different types of vector multiplication. The dot product (A · B) returns a scalar value equal to |A||B|cos(θ) and measures how parallel two vectors are. The cross product (A × B) returns a vector perpendicular to both inputs with magnitude |A||B|sin(θ) and measures how perpendicular two vectors are. The dot product is commutative (A · B = B · A), while the cross product is anti-commutative (A × B = -B × A). The dot product works in any dimension; the cross product is defined only in 3D (and 7D).
What does the cross product represent geometrically?
Geometrically, the cross product A × B represents a vector perpendicular to the plane formed by A and B. Its magnitude equals the area of the parallelogram spanned by A and B. The direction follows the right-hand rule: point your fingers along A, curl them toward B, and your thumb points in the direction of A × B. If the cross product is the zero vector, the two input vectors are parallel (or one is the zero vector). This geometric interpretation is widely used for finding surface normals, computing torque, and determining oriented areas.
Can you compute cross product in 2D?
Strictly speaking, the cross product is only defined in three dimensions. However, for 2D vectors A = (a₁, a₂) and B = (b₁, b₂), you can compute a pseudo cross product by treating them as 3D vectors with zero z-components: A = (a₁, a₂, 0) and B = (b₁, b₂, 0). The result is (0, 0, a₁b₂ - a₂b₁), a vector pointing along the z-axis. The scalar value a₁b₂ - a₂b₁ gives the signed area of the parallelogram formed by the two vectors and is useful for determining if B is clockwise or counterclockwise relative to A.
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