Understanding the 3D Cross Product

The cross product is a fundamental operation in vector mathematics that produces a vector perpendicular to two input vectors. It's crucial in physics for calculating torque, in computer graphics for surface normals, and in engineering for moment calculations.

Calculation Steps

Identify components of Vector A (Ax, Ay, Az) and Vector B (Bx, By, Bz)
Calculate X-component: (Ay × Bz) - (Az × By)
Calculate Y-component: (Az × Bx) - (Ax × Bz)
Calculate Z-component: (Ax × By) - (Ay × Bx)
The result is a new vector (Cx, Cy, Cz) perpendicular to both A and B

Practical Example

For vectors A(2,3,4) and B(5,6,7):

Cx = (3×7) - (4×6) = 21 - 24 = -3
Cy = (4×5) - (2×7) = 20 - 14 = 6
Cz = (2×6) - (3×5) = 12 - 15 = -3

Resulting vector: (-3, 6, -3)

Important Notes

The cross product is anti-commutative: A × B = - (B × A)
The magnitude of the result equals |A| × |B| × sin(θ)
Parallel vectors yield a zero cross product

Cross Product Calculator - 3D Vector Product

Vector A

Vector B

Cross Product Result

Understanding the 3D Cross Product

Calculation Steps

Practical Example

Important Notes

Vector Magnitude Calculator

Frequently Asked Questions

What does the cross product represent physically?

Why is the cross product anti-commutative?

Can I calculate cross product for 2D vectors?

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Cross Product Calculator - 3D Vector Product

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