MathIsimple
Vector Algebra

Cross Product Calculator

Compute 3D vector cross product, magnitude, and direction with step-by-step determinant expansion.

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Cross Product
Enter components for vectors a and b

Enter the first 3D vector components (e.g., 1, 2, 3)

Enter the second 3D vector components (e.g., 4, 5, 6)

What Is the Cross Product?

The cross product of two 3D vectors is a vector perpendicular to both. It differs from the dot product, which returns a scalar.

Geometric meaning: its magnitude equals the area of the parallelogram spanned by the two vectors; direction is orthogonal to their plane, determined by the right-hand rule.

Notation: a⃗×b⃗\vec{a} \times \vec{b} ("a cross b")
Cross Product Formula & Calculation

Using the 3×3 determinant with unit vectors:

∣ijkaxayazbxbybz∣=i(aybz−azby)−j(axbz−azbx)+k(axby−aybx) \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \mathbf{i}(a_y b_z - a_z b_y) - \mathbf{j}(a_x b_z - a_z b_x) + \mathbf{k}(a_x b_y - a_y b_x)

Component formulas:

x=aybz−azby,y=azbx−axbz,z=axby−aybxx = a_y b_z - a_z b_y, \quad y = a_z b_x - a_x b_z, \quad z = a_x b_y - a_y b_x
Tip: Anti-commutative: a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a}).
Properties of the Cross Product
  • Anti-commutative: a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})
  • Distributive: a⃗×(b⃗+c⃗)=a⃗×b⃗+a⃗×c⃗\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
  • Collinearity: if a⃗∥b⃗\vec{a} \parallel \vec{b} then a⃗×b⃗=0⃗\vec{a} \times \vec{b} = \vec{0} (zero vector)
  • Scalar multiplication: (ka⃗)×b⃗=k(a⃗×b⃗)(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})
Cross vs Dot: Cross product yields a vector; dot product yields a scalar.
Applications of the Cross Product
  • Torque: τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}
  • Lorentz force: F⃗=qv⃗×B⃗\vec{F} = q \vec{v} \times \vec{B}
  • Angular momentum: L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}
  • 3D modeling: surface normal calculation
  • Mechanics: rotation direction and force relations
Examples
Click to auto-fill
Example
(1,2,3) × (4,5,6)
Example
(2,0,0) × (0,2,0)
Example
Parallel vectors
Example
k × i