✕ Cross Product Calculator

Cross Product Calculator

The cross product of two 3D vectors produces a vector perpendicular to both, and this calculator computes it using the determinant method with i, j, k components. Shows determinant expansion, i/j/k components, result vector, magnitude, and verifies a·(a×b) = 0.

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Cross Product a × b

3D vectors only — i, j, k components

✓ Step-by-Step✓ Magnitude✓ Free⚡ Loading precision engine…
Vector a = (a₁, a₂, a₃)
×
Vector b = (b₁, b₂, b₃)
Quick Examples

Cross Product Formula

The cross product of two 3D vectors a and b is a vector perpendicular to both:

$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix} = (a_2b_3-a_3b_2)\mathbf{i} - (a_1b_3-a_3b_1)\mathbf{j} + (a_1b_2-a_2b_1)\mathbf{k}$$

Key Properties

  • a × b is perpendicular to both a and b
  • |a × b| = |a||b|sin(θ) = area of parallelogram formed by a and b
  • a × a = 0 (cross product with itself is zero)
  • a × b = −(b × a) (anti-commutative)
  • a · (a × b) = 0 (perpendicularity confirmed)

Right-Hand Rule

Point your right hand fingers from a toward b. Your thumb points in the direction of a × b.

Worked Example

For a = (1,0,0) and b = (0,1,0): the cross product a × b = (0,0,1). Notice the result is perpendicular to both original vectors — this is always true for a cross product.

Why Only 3D Vectors?

The cross product, as calculated here, is only mathematically defined for vectors with exactly 3 components. This isn't a limitation of this tool specifically — it's how the operation is defined in standard vector algebra.

Applications

  • Finding a vector perpendicular to a surface (used constantly in 3D graphics and lighting)
  • Calculating torque and angular momentum in physics
  • Determining the direction of rotation in mechanics

Frequently Asked Questions

The cross product is defined specifically for 3-dimensional space as part of its mathematical definition. There isn't a standard 2D or 4D version used in typical vector algebra.
Yes — a × b and b × a point in exactly opposite directions. Swapping the order flips the sign of the result.
The result is a new vector that's perpendicular to both original vectors, with a length related to the area of the parallelogram they form.

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