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Vector Cross Product

Generate perfect perpendicular 3D surface normals out of flat 2D geometries.

How does it work? ↓
Vector A × Vector B = Vector C

Calculated Matrix

Vector A: [0, 0, 0]
Vector B: [0, 0, 0]
Cross C: [0, 0, 0]

Vector A (Flat Ground)

Vector B (Flat Ground)

Code Snippet

Related Tools

The 3D Cross Product

If the Dot Product tells you how similar two vectors are (returning a single floating number), the Cross Product calculates a mathematically pure 90-degree angle (returning a brand new Vector).

When you feed the mathematical matrix two vectors, it evaluates exactly which direction is perfectly perpendicular to both inputs.

Why restrict the canvas to Z=0?

To fundamentally grasp the algebraic output of the Cross Product, it's easiest to lock Inputs A and B entirely flat to the ground (where Z equals 0). By moving Vectors A and B around the flat X/Y grid, the mathematics dictate that the ONLY possible direction perfectly perpendicular to a generic flat plane is straight up into the Z-Axis sky.

Vector C.z = (A.x * B.y) - (A.y * B.x)

Game Engines Output

If you switch the dropdown to Godot or Unity below, notice that you don't actually need to script the literal math yourself! Modern engines contain a `Vector3.Cross()` wrapper that natively computes the 3x3 Determinant scalar for you. This generates your physics surface normals to bounce lasers and compute polygon lighting!

Answers to Common Developer Questions

What does the Vector Cross Product actually do?
The Cross Product takes two distinct Vectors (A and B) and calculates a brand new Vector (C) that is **perfectly perpendicular** (90 degrees) to both of them. In game development, this is how you mathematically generate 3D 'Surface Normals' for lighting and physics engines.
How do you calculate a Cross Product?
Unlike the Dot Product (which outputs a single standard number), the Cross Product outputs an entirely new 3D Vector. The 3x3 Determinant formula is: `C.x = A.y*B.z - A.z*B.y`, `C.y = A.z*B.x - A.x*B.z`, and `C.z = A.x*B.y - A.y*B.x`.
Why does multiplying a 2D plane result in the Z-Axis?
By definition, the output Vector is perpendicular to the inputs! If you have Vector A and Vector B both lying completely flat on the ground (The X and Y 2D axes), the only possible direction that is exactly 90 degrees away from both of them is point straight UP into the sky (The Z-Axis!).