Use a Cross Product calculator to compute the cross product of two 3D vectors and get the resulting perpendicular vector. It also returns the magnitude, which equals the area of the parallelogram formed by the two vectors.
- Enter vector A components (Ax, Ay, Az).
- Enter vector B components (Bx, By, Bz).
- Choose units for the input vectors (optional, affects labeling).
- Click Calculate to get A × B and its magnitude.
- Click Reset to clear all fields.
What a Cross Product calculator computes
The cross product of two vectors A and B (in 3D) is written as A × B. The result is a new vector that is perpendicular to both inputs, following the right-hand rule.
A Cross Product calculator computes the three components of the cross product and the magnitude (length) of the resulting vector.
The core formula (vector components)
Let A = (Ax, Ay, Az) and B = (Bx, By, Bz). The cross product is:
| Component | Formula |
|---|---|
| (A × B)x | AyBz − AzBy |
| (A × B)y | AzBx − AxBz |
| (A × B)z | AxBy − AyBx |
These component formulas come directly from the determinant form of the cross product.
Magnitude and geometric meaning
The magnitude of the cross product is:
|A × B| = |A| |B| sin(θ)
Here, θ is the angle between the vectors. Geometrically, |A × B| equals the area of the parallelogram formed by A and B.
If the vectors are parallel (θ = 0°) or anti-parallel (θ = 180°), then sin(θ) = 0, so the cross product becomes the zero vector.
Units and what they mean
The calculator can label your input units, but the math is the same regardless of units. If your vectors represent lengths (like meters), then the cross product magnitude has units of square meters because it corresponds to an area.
If your vectors represent forces (newtons), the cross product magnitude has units of newton-meters, which relates to rotational effects (torque-like quantities).
How to interpret the direction (right-hand rule)
The cross product direction follows the right-hand rule: point your index finger along A, your middle finger along B. Your thumb points in the direction of A × B.
Swapping the order changes the sign: A × B = −(B × A). So if you reverse the inputs, the magnitude stays the same but the resulting vector flips direction.
Practical examples
Example 1: Find the perpendicular direction for a plane
Suppose a plane is defined by two direction vectors: A = (2, 0, 1) and B = (0, 3, 4). The cross product A × B gives a vector perpendicular to both, which can be used as a plane normal.
The calculator will output the normal vector components and its magnitude. If the magnitude is near zero, the vectors are nearly parallel, meaning they don’t define a strong plane direction.
Example 2: Compute parallelogram area from two vectors
Let A = (3, 1, 0) and B = (1, 2, 0). Even though both vectors lie in the same plane, the cross product magnitude still gives the parallelogram area.
Use the calculator’s magnitude output to get the area directly. This is common in geometry, graphics, and physics problems involving surface area.
Common mistakes to avoid
- Mixing up components: double-check which value goes into x, y, and z.
- Forgetting order: A × B is not the same as B × A.
- Assuming “magnitude” is always non-zero: parallel vectors produce a zero cross product.
- Ignoring units: magnitude corresponds to an area-like quantity when vectors represent lengths.
Frequently Asked Questions
What is a Cross Product calculator used for?
A Cross Product calculator computes the vector result of A × B for two 3D vectors. It returns the three cross-product components and the magnitude |A × B|. The direction is perpendicular to both inputs, and the magnitude equals the parallelogram area formed by the vectors.
How do I know if two vectors are parallel using the result?
If the cross product magnitude is zero (or extremely close to zero with rounding), the vectors are parallel or anti-parallel. In that case, the cross-product components should all be zero. This happens because sin(θ) becomes zero when θ is 0° or 180°.
Why does swapping A and B change the answer?
Cross products are order-dependent: A × B = −(B × A). That means the magnitude stays the same, but the vector direction flips. In component form, each cross-product component changes sign when you reverse the order of the vectors.
What units do I get from the cross product magnitude?
Units depend on how you define the vectors. If A and B are in meters, then |A × B| behaves like an area and is in square meters. If they are forces, the magnitude follows the product of units (often linked to rotational effects).
Can the cross product be computed for 2D vectors?
The standard cross product is defined for 3D vectors. For 2D vectors, you typically embed them in 3D by adding a zero z-component: (x, y) becomes (x, y, 0). The cross product then points along the z-axis, and its magnitude matches the 2D area.