cross product calculator

What is the cross product?

The cross product is an operation between two vectors in 3D space that produces a new vector perpendicular to both original vectors. If you have vectors A and B, then A × B points in a direction given by the right-hand rule.

Unlike the dot product (which returns a scalar), the cross product returns a vector. This makes it especially useful in physics, engineering, computer graphics, robotics, and geometry.

How to use this cross product calculator

1) Enter vector components

Fill in Ax, Ay, Az for vector A and Bx, By, Bz for vector B. Decimal values and negative values are both supported.

2) Click “Calculate Cross Product”

The calculator computes each component using the standard determinant expansion:

• x-component: AyBz − AzBy
• y-component: AzBx − AxBz
• z-component: AxBy − AyBx

3) Read the result

You will see the resulting vector, its magnitude, and a quick note about whether the input vectors appear parallel (a zero or near-zero cross product).

Geometric meaning

The magnitude of the cross product, |A × B|, equals the area of the parallelogram spanned by A and B:

|A × B| = |A||B|sin(θ)

This means the cross product captures both orientation and “spread” between vectors. If vectors are parallel, sin(θ)=0, so the cross product is zero.

Why order matters

Cross product is anti-commutative:

A × B = −(B × A)

Switching the order flips the direction of the resulting vector. This is a common source of mistakes in torque and normal-vector calculations.

Common applications

• Physics: torque (τ = r × F), angular momentum (L = r × p), magnetic force direction.
• Computer graphics: surface normals for lighting, backface culling, orientation tests.
• Engineering: moment calculations and 3D rigid-body mechanics.
• Geometry: area calculations and tests for collinearity/parallelism in 3D.

Worked example

Let A = (2, 3, 4) and B = (5, 6, 7). Then:

• x = 3·7 − 4·6 = 21 − 24 = −3
• y = 4·5 − 2·7 = 20 − 14 = 6
• z = 2·6 − 3·5 = 12 − 15 = −3

A × B = (−3, 6, −3)

Try this quickly with the “Load Example” button in the calculator above.

Common mistakes to avoid

• Using 2D vectors directly without embedding in 3D (z = 0).
• Mixing up the middle component sign during expansion.
• Assuming A × B = B × A (it is not).
• Forgetting units in physics contexts (e.g., N·m for torque).

Quick identity list

• A × A = (0, 0, 0)
• A × B = −B × A
• (kA) × B = k(A × B)
• A × (B + C) = A × B + A × C

Final thoughts

A reliable cross product calculator is a practical tool whenever direction, orientation, and area matter in 3D. Use this page to verify hand calculations, speed up homework checks, or support technical workflows in physics, engineering, and graphics.