-3
6
-3
7.3485
7.3485
sq units
3.6742
sq units
56
5.3852
10.4881
-3
6
-3
7.3485
7.3485
sq units
3.6742
sq units
56
5.3852
10.4881
The Cross Product Calculator computes the vector product of two three-dimensional vectors. Unlike the dot product, which yields a scalar, the cross product produces a new vector that is perpendicular to both input vectors. This operation is fundamental to physics, engineering, and computer graphics, where it is used to find normal vectors, compute torques, and determine the orientation of surfaces.
Given vectors A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂), their cross product A × B is a vector whose direction is determined by the right-hand rule: if you curl the fingers of your right hand from A toward B, your thumb points in the direction of A × B. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, providing a direct geometric interpretation.
The cross product is exclusively a three-dimensional operation. While the dot product generalizes naturally to any number of dimensions, the cross product as traditionally defined only works in 3D (and, through a different formalism, in 7D). This makes it a uniquely spatial operation, intimately connected to the geometry of three-dimensional Euclidean space and the structure of the rotation group SO(3).
In physics, the cross product appears in the definition of torque (τ = r × F), the Lorentz force on a charged particle (F = qv × B), angular momentum (L = r × p), and the Poynting vector (S = E × B/μ₀) in electromagnetism. Each of these involves finding a quantity perpendicular to two given directions, which is precisely what the cross product computes.
In computer graphics, the cross product is used to compute surface normals for lighting calculations. Given two edge vectors of a triangle, their cross product yields the normal to the triangle's surface, which is then used in shading models to determine how light reflects off the surface. This single operation is performed billions of times per second in modern graphics pipelines.
This calculator accepts two 3D vectors and computes the cross product vector, its magnitude, and the area of the parallelogram spanned by the input vectors. It is an essential tool for anyone working with spatial geometry, rotational mechanics, or 3D rendering.
The cross product is computed using the determinant formula:
$$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix}$$
Expanding the determinant:
$$\mathbf{A} \times \mathbf{B} = (y_1 z_2 - z_1 y_2)\,\hat{\mathbf{i}} - (x_1 z_2 - z_1 x_2)\,\hat{\mathbf{j}} + (x_1 y_2 - y_1 x_2)\,\hat{\mathbf{k}}$$
Which can be written component-wise as:
$$(\mathbf{A} \times \mathbf{B})_x = y_1 z_2 - z_1 y_2$$
$$(\mathbf{A} \times \mathbf{B})_y = z_1 x_2 - x_1 z_2$$
$$(\mathbf{A} \times \mathbf{B})_z = x_1 y_2 - y_1 x_2$$
The magnitude of the cross product equals:
$$|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}|\,|\mathbf{B}|\,\sin \theta$$
This magnitude also equals the area of the parallelogram formed by vectors A and B.
The cross product vector is perpendicular to both input vectors. Its direction follows the right-hand rule. The magnitude equals |A||B|sin θ, which is maximized when the vectors are perpendicular (θ = 90°) and zero when they are parallel (θ = 0° or 180°). The parallelogram area equals the magnitude and represents the area enclosed by the two vectors when placed tail-to-tail. Half this value gives the area of the triangle formed by the two vectors.
Inputs
Results
X: (3)(7) − (4)(6) = 21 − 24 = −3. Y: (4)(5) − (2)(7) = 20 − 14 = 6. Z: (2)(6) − (3)(5) = 12 − 15 = −3. Magnitude = √(9 + 36 + 9) = √54 ≈ 7.3485.
Inputs
Results
î × ĵ = k̂. The cross product of the standard basis vectors x and y produces the z basis vector, confirming the right-hand rule. The parallelogram area is 1 (unit square).
No. The cross product is anti-commutative: A × B = −(B × A). Reversing the order reverses the direction of the result vector. This is because the right-hand rule gives opposite directions when the operands are swapped.
When vectors are parallel (or anti-parallel), their cross product is the zero vector (0, 0, 0). This is because sin 0° = sin 180° = 0, so the magnitude |A||B|sin θ = 0. Parallel vectors span no area.
The cross product as traditionally defined is unique to 3D because it relies on the existence of a unique perpendicular direction to any plane. In 2D, there's only one perpendicular direction (making it a scalar). In higher dimensions, the perpendicular space to a plane is multi-dimensional. The mathematical structure behind this is related to the Hodge dual and exterior algebra.
Given a triangle with vertices P₁, P₂, P₃, compute edge vectors e₁ = P₂ − P₁ and e₂ = P₃ − P₁. Their cross product e₁ × e₂ is perpendicular to the triangle's surface. Normalizing this gives the unit normal used in lighting calculations.
Torque is defined as τ = r × F, where r is the position vector from the pivot to the point of force application, and F is the force vector. The cross product naturally captures both the perpendicular component of force and the lever arm distance.
Yes. The area of a triangle formed by vectors A and B is exactly half the magnitude of their cross product: Area = |A × B| / 2. This is because a triangle is half a parallelogram.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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