Калькулятор векторного произведения

What the cross product gives you

The cross product a × b of two 3D vectors is a third vector that's perpendicular to both a and b, with a magnitude equal to the area of the parallelogram they span. It shows up constantly in physics and 3D graphics: computing a surface normal, torque, angular momentum, or the area of a triangle defined by three points.

The formula used

For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):

a × b = (a₂b₃ − a₃b₂,  a₃b₁ − a₁b₃,  a₁b₂ − a₂b₁)

From there, the calculator also derives:

• Magnitude of a, b, and a × b (via √(x² + y² + z²))
• Unit cross product — a × b divided by its own magnitude, giving a direction with no length
• Parallelogram area — equal to |a × b|
• Triangle area — half the parallelogram area, useful for the area of a triangle with two sides a and b
• Dot product, used to get the angle between a and b via cos θ = (a · b) / (|a||b|)
• A perpendicularity check — a · (a × b) and b · (a × b), which are always 0 in exact arithmetic and confirm the cross product really is perpendicular to both inputs

The right-hand rule

The direction of a × b (not just its components) follows the right-hand rule: point your fingers along a, curl them toward b, and your thumb points along a × b. Swapping the order flips the sign — b × a = −(a × b) — which is why cross product order matters and, unlike the dot product, isn't commutative.

When is the cross product the zero vector?

Whenever a and b are parallel (including when one is the zero vector) — there's no unique perpendicular direction to point in, and the "area" they span collapses to zero. The calculator flags this explicitly rather than showing a division-by-zero unit vector.

How to use it

1. Enter the x, y, z components of vector a.
2. Enter the x, y, z components of vector b.
3. Read the cross product vector, its magnitude/unit form, the areas it implies, and the angle between the two original vectors.

FAQ

Can I use this for 2D vectors? Yes — leave the z component at 0 for both vectors. The result's x and y components will also be 0, leaving only a z component, which is the standard "2D cross product" (a scalar) most people actually want.

Why is my cross product's z component nonzero when I only entered x, y values? Check that you actually left both z fields at 0 — if either vector has a nonzero z, the vectors aren't coplanar with the xy-plane and the cross product won't be purely vertical.

Does this calculator store my data? No. Everything runs in your browser — your vectors and results are never sent to or stored on our servers.