Compute the cross product of two 3D vectors, find the parallelogram area, unit normal direction, angle between vectors, and scalar triple product with step-by-step breakdowns.
The cross product (or vector product) is a binary operation on two 3D vectors that produces a third vector perpendicular to both inputs. If a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the cross product a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Unlike the dot product, the result is a vector, not a scalar, and it is defined only in three dimensions (with a special extension to seven dimensions).
The magnitude ‖a × b‖ equals the area of the parallelogram spanned by the two vectors, and its direction follows the right-hand rule: curl the fingers of your right hand from a toward b and your thumb points in the direction of a × b. This geometric meaning makes the cross product indispensable in physics (torque, angular momentum, magnetic force), computer graphics (surface normals, lighting calculations), and engineering (moment of a force).
This calculator takes two 3D vectors and computes the full cross product vector, its magnitude (parallelogram area), the unit normal direction, the angle between the input vectors, and a scalar triple product for volume calculations. Six presets cover classic cases like i × j and arbitrary vectors. A step-by-step computation table shows exactly how each component is derived, and bar charts visualize the result components. A properties table at the bottom summarizes the key algebraic rules: anti-commutativity, distributivity, and the crucial perpendicularity guarantee.
The cross product is essential for computing surface normals in 3D rendering, calculating torque in mechanics, finding the area of triangles in computational geometry, and testing whether two vectors are parallel (a × b = 0 ⟺ a ∥ b).
The cross product formula involves six multiplications and three subtractions with careful sign handling (the j-component is negated) — errors in sign or component pairing are extremely common. This calculator computes the full result vector, its magnitude (parallelogram area), the unit normal direction, the angle between inputs, and the scalar triple product, all in one step. It is essential for physics students computing torque or angular momentum, 3D graphics programmers finding surface normals, and engineers calculating moments of force.
a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k; ‖a × b‖ = ‖a‖ ‖b‖ sin θ
Result: a × b = (−3, 6, −3)
i: 2×6 − 3×5 = −3. j: 3×4 − 1×6 = 6. k: 1×5 − 2×4 = −3. ‖a × b‖ = √(9+36+9) = √54 ≈ 7.348. This is the area of the parallelogram spanned by a and b.
The cross product a × b is computed using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row: a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|. Expanding along the first row gives the components: (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k. The direction follows the **right-hand rule**: point your fingers along a, curl them toward b, and your thumb points in the direction of a × b. Reversing the order gives b × a = −(a × b), which is why the cross product is anti-commutative.
The magnitude ‖a × b‖ = ‖a‖ ‖b‖ sin θ equals the area of the parallelogram spanned by a and b. Half this value gives the triangle area, making the cross product a key tool in computational geometry. The result vector is always perpendicular to both inputs, which is why it is used to compute surface normals in 3D rendering: given two edge vectors of a triangle, their cross product yields the face normal for lighting calculations. When a × b = 0, the vectors are parallel (sin θ = 0).
In mechanics, **torque** τ = r × F is the cross product of the position vector and force, giving the rotational effect. **Angular momentum** L = r × p uses the same structure. In electromagnetism, the **Lorentz force** on a charged particle is F = qv × B, where v is velocity and B is the magnetic field. The **scalar triple product** a · (b × c) gives the signed volume of the parallelepiped formed by three vectors, and is zero when they are coplanar. These applications make the cross product one of the most frequently used operations in applied mathematics and physics.
The cross product of two 3D vectors produces a new vector perpendicular to both inputs. Its magnitude equals the parallelogram area and its direction follows the right-hand rule.
A unique perpendicular direction to two vectors exists only in 3D (and 7D by algebraic extension). In 2D, a pseudo-scalar substitute (a₁b₂ − a₂b₁) gives the signed area.
It equals the area of the parallelogram formed by the two vectors. Half the magnitude gives the area of the triangle they span.
Point your right-hand fingers along vector a, curl them toward b — your thumb points in the direction of a × b. Use this as a practical reminder before finalizing the result.
When the two vectors are parallel or anti-parallel (or one is the zero vector), because sin 0° = sin 180° = 0. Keep this note short and outcome-focused for reuse.
a · (b × c) is a scalar whose absolute value equals the volume of the parallelepiped spanned by a, b, and c. It is zero when the three vectors are coplanar.