Cross Product Calculator
Find the cross product a × b of two three-dimensional vectors, giving a new vector perpendicular to both along with its magnitude.
Cross product a × b = (0, 0, 1)
- 1
x-component: a₂b₃ − a₃b₂
0 × 0 − 0 × 1 = 0 - 2
y-component: a₃b₁ − a₁b₃
0 × 0 − 1 × 0 = 0 - 3
z-component: a₁b₂ − a₂b₁
1 × 1 − 0 × 0 = 1 - 4
Magnitude |a × b|
√(0² + 0² + 1²) = 1Equals the area of the parallelogram spanned by a and b.
How does this calculator work?
The cross product a × b of two 3D vectors is (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), a vector perpendicular to both. Enter three components for each vector and the calculator returns that vector plus its magnitude, which equals the area of the parallelogram the two vectors span.
Formula
How this is calculated
Enter two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), each with exactly three components separated by commas or spaces. The calculator parses both lists and requires three finite numbers in each before computing.
The cross product is computed component by component: the x-component is a₂b₃ − a₃b₂, the y-component is a₃b₁ − a₁b₃, and the z-component is a₁b₂ − a₂b₁. The result is a vector that is perpendicular to both a and b, following the right-hand rule. Its magnitude equals |a||b|sin(θ), where θ is the angle between the inputs, and equals the square root of the sum of the squared components.
The magnitude also represents the area of the parallelogram spanned by a and b. If the two vectors are parallel (or either is the zero vector), the cross product is the zero vector and the magnitude is 0. The cross product is defined only in three dimensions, so each input must have precisely three components.
Frequently asked questions
It equals |a||b|sin(θ), the area of the parallelogram formed by the two vectors. It is zero when the vectors are parallel or one of them is the zero vector.
No. The cross product is anticommutative: a × b = −(b × a). Swapping the order of the vectors reverses the direction of the resulting vector while keeping its magnitude the same.
The cross product as defined here only exists in three dimensions. Providing fewer or more than three components is rejected, so enter precisely three numbers per vector.
Also known as
TG we-Calculate Editorial Team. (2026). Cross Product Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cross-product-calculator
TG we-Calculate Editorial Team. "Cross Product Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/cross-product-calculator.
TG we-Calculate Editorial Team, "Cross Product Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cross-product-calculator
@misc{wecalculate_cross_product_calculator, title = {Cross Product Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cross-product-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?