Cross Product Calculator
Calculate the cross product (vector product) of two 3D vectors with detailed step-by-step solutions. Visualize the resulting perpendicular vector in interactive 3D space and understand the geometric relationships between vectors with our comprehensive cross product calculator.
Vector Input (3D Only)
Cross Product Theory
What is the Cross Product?
The cross product (also called vector product) is an operation on two 3D vectors that produces a third vector perpendicular to both input vectors. Unlike the dot product which returns a scalar, the cross product returns a vector that represents the axis of rotation and has magnitude equal to the area of the parallelogram formed by the two vectors.
Cross Product Formula:
A × B = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]
Determinant Form:
A × B = |i j k |
|a₁ a₂ a₃|
|b₁ b₂ b₃|
Magnitude: |A × B| = |A| × |B| × sin(θ)
Calculation Steps
Step 1: Set up the determinant
Arrange vectors in a 3×3 matrix with unit vectors i, j, k in the first row
Step 2: Calculate i component
i(a₂b₃ - a₃b₂) - expand along first column
Step 3: Calculate j component
-j(a₁b₃ - a₃b₁) - note the negative sign
Step 4: Calculate k component
k(a₁b₂ - a₂b₁) - complete the determinant expansion
Key Properties
Anti-commutative
A × B = -(B × A)
Distributive
A × (B + C) = A × B + A × C
Scalar Multiplication
(kA) × B = k(A × B)
Perpendicularity
(A × B) ⊥ A and (A × B) ⊥ B
Zero Vector
A × A = 0
Right-Hand Rule
Direction follows right-hand rule
Applications
Physics
Torque calculation, angular momentum, magnetic force
Computer Graphics
Surface normals, lighting calculations, 3D rotations
Engineering
Moment calculations, structural analysis, robotics
Geometry
Area calculations, plane equations, perpendicular vectors
Navigation
Course corrections, orientation calculations
Game Development
Collision detection, camera controls, physics engines
Worked Examples
Example 1: Basic Cross Product
Problem:
Calculate A × B where A = [1, 0, 0] and B = [0, 1, 0]
Solution:
Step 1: Set up determinant with i, j, k
Step 2: i component: (0)(0) - (0)(1) = 0
Step 3: j component: -[(1)(0) - (0)(0)] = 0
Step 4: k component: (1)(1) - (0)(0) = 1
Result: A × B = [0, 0, 1] (unit vector in z direction)
Example 2: General Cross Product
Problem:
Calculate A × B where A = [2, 1, 3] and B = [1, 4, 2]
Solution:
Step 1: i component: (1)(2) - (3)(4) = 2 - 12 = -10
Step 2: j component: -[(2)(2) - (3)(1)] = -(4 - 3) = -1
Step 3: k component: (2)(4) - (1)(1) = 8 - 1 = 7
Result: A × B = [-10, -1, 7]
Magnitude: |A × B| = √(100 + 1 + 49) = √150 ≈ 12.25
Example 3: Parallel Vectors
Problem:
Calculate A × B where A = [2, 4, 6] and B = [1, 2, 3]
Solution:
Step 1: Notice that A = 2B (parallel vectors)
Step 2: i component: (4)(3) - (6)(2) = 12 - 12 = 0
Step 3: j component: -[(2)(3) - (6)(1)] = -(6 - 6) = 0
Step 4: k component: (2)(2) - (4)(1) = 4 - 4 = 0
Result: A × B = [0, 0, 0] (parallel vectors have zero cross product)
Frequently Asked Questions
The cross product as commonly defined only works in 3D because it relies on the specific properties of 3D space. While generalizations exist (like the wedge product in higher dimensions), the traditional cross product that produces a perpendicular vector only makes sense in 3D space.
Cross product returns a vector perpendicular to both input vectors, while dot product returns a scalar. Cross product magnitude equals the area of the parallelogram formed by the vectors, while dot product measures how much vectors point in the same direction.
When A × B = 0, the vectors are parallel (or anti-parallel). This happens because parallel vectors don't define a unique perpendicular direction, and the area of the "parallelogram" they form is zero.
Point your right hand fingers in the direction of the first vector (A), curl them toward the second vector (B), and your thumb points in the direction of A × B. This determines the orientation of the cross product vector.
A × B = -(B × A) because swapping the order of vectors reverses the direction according to the right-hand rule. The magnitude stays the same, but the direction flips. This is different from regular multiplication which is commutative.
Cross products are essential for calculating surface normals (for lighting), determining face orientation (front vs back), computing rotations, and many collision detection algorithms. They're fundamental to 3D graphics programming.