How do I use the NexTools Cross Product Calculator - 3D Vector (A × B) & Normal Vector?

Simply open the Cross Product Calculator - 3D Vector (A × B) & Normal Vector on NexTools, enter your values or parameters in the input fields, and get instant, accurate results directly in your web browser.

Is the Cross Product Calculator - 3D Vector (A × B) & Normal Vector tool completely free to use?

Yes, Cross Product Calculator - 3D Vector (A × B) & Normal Vector on NexTools is 100% free with unlimited usage, zero subscriptions, and no registration or API keys required.

Does NexTools store my data when using Cross Product Calculator - 3D Vector (A × B) & Normal Vector?

No. All calculations and operations are processed securely within your browser session or secure server requests. We never store or share your input data.

Can I use Cross Product Calculator - 3D Vector (A × B) & Normal Vector on mobile devices?

Yes, Cross Product Calculator - 3D Vector (A × B) & Normal Vector is fully responsive and optimized to work seamlessly across mobile phones, tablets, and desktop computers.

What is the cross product of two vectors?

The cross product (or vector product) of two 3D vectors A and B produces a third vector (A × B) that is perpendicular to both original vectors, satisfying the right-hand rule.

What is the formula for the cross product?

A × B = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k.

Is the cross product commutative?

No, the cross product is anti-commutative: A × B = -(B × A). Swapping the order flips the direction of the resulting vector by 180 degrees.

What does a cross product of zero mean?

If A × B = 0 for non-zero vectors, it proves that the vectors are parallel or collinear (angle θ = 0° or 180°).

3D Spatial Geometry

3D Vector Cross Product (A × B) Calculator

Compute orthogonal cross products, normal vectors, enclosed parallelogram areas, and step-by-step 3x3 determinant solutions.

Vector A (x, y, z)

Vector B (x, y, z)

Cross Product Vector (A × B)

(-3, 6, -3)

-3i + 6j - 3k

Magnitude |A × B|

7.3485

Parallelogram Area

7.3485 sq units

Triangle Area

3.6742 sq units

• Unit Normal Vector (n̂): (-0.4082, 0.8165, -0.4082)

• Anti-Commutative Product (B × A): (3, -6, 3)

• Determinant Formula: i(3×7 - 4×6) - j(2×7 - 4×5) + k(2×6 - 3×5)

Understanding the 3D Vector Cross Product

Unlike the dot product which yields a scalar number, the cross product of two vectors produces a new vector oriented perpendicular (normal) to the plane containing the original vectors. The direction follows the Right-Hand Rule: pointing your index finger along Vector A and middle finger along Vector B results in your thumb pointing directly along Vector A × B.

Frequently Asked Questions

Why does A × B equal -(B × A)?

The cross product is anti-commutative. Reversing the input vectors reverses the right-hand rule orientation, pointing the resulting vector in the exact opposite direction.

How does the cross product calculate parallelogram area?

The magnitude |A × B| is geometrically equal to |A| |B| sin(θ), which is precisely the geometric formula for the area of the parallelogram formed by vectors A and B.