Cross Product Calculator
The definitive 3D vector cross product calculator for computing vector products (a × b), orthogonal normal vectors, 3x3 determinant expansions, and spatial parallelogram areas.
3D Isometric Projection & Parallelogram Area
(Right-Hand Rule)3x3 Determinant & Sarrus Step-by-Step Expansion Full Algebraic Proof
How to Calculate the Cross Product of Two 3D Vectors
The cross product of two 3D vectors a = ⟨a_x, a_y, a_z⟩ and b = ⟨b_x, b_y, b_z⟩ yields a third vector perpendicular to both, calculated via 3x3 determinant cofactor expansion: a × b = ⟨(a_y b_z - a_z b_y), -(a_x b_z - a_z b_x), (a_x b_y - a_y b_x)⟩. The magnitude ||a × b|| equals the area of the spanned parallelogram: ||a|| ||b|| sin(θ).
Anatomy of the 3D Vector Cross Product
The cross product (or vector product) is a mathematical operation defined exclusively in three-dimensional space (ℝ³) that combines two vectors to produce a third vector orthogonal (perpendicular) to both.
While the dot product measures collinearity and alignment, the cross product measures orthogonality, rotational influence, and enclosed spatial area.
Reversing input order flips the direction of the normal vector 180°.
The cross product is perpendicular to both original vectors simultaneously.
The cross product of any vector with itself or a scalar multiple is zero.
The 3x3 Determinant Expansion & Sarrus Rule
The most reliable method to compute a cross product by hand is using a pseudo-determinant of a 3×3 matrix constructed with standard unit basis vectors i, j, k:
| ax ay az | = i(aybz − azby) − j(axbz − azbx) + k(axby − aybx)
| bx by bz |
Notice the negative sign attached to the j cofactor term. Expanding the negative sign into the parenthesis gives +(azbx - axbz)j.
Right-Hand Rule & Geometric Parallelogram Area
The geometric interpretation of the cross product combines vector length, angle, and 3D spatial orientation:
The Right-Hand Rule
Point the fingers of your right hand along vector a, and curl them toward vector b through the smallest angle θ. Your extended thumb points in the exact direction of a × b.
Parallelogram & Triangle Area
The magnitude ||a × b|| = ||a|| ||b|| sin(θ) equals the exact surface area of the parallelogram bounded by a and b. The area of the triangle formed by connecting their tips is 0.5 × ||a × b||.
Physical Applications: Torque, Angular Momentum & Lorentz Force
Torque & Rotational Force
The rotational twist generated by a force on a lever arm: τ = r × F, measured in Newton-meters (N·m).
Magnetic Lorentz Force
The deflecting force on an electric charge moving through a magnetic field: F = q(v × B).
3D Game Engine Polygon Normals
Graphics pipelines compute face normal vectors from three triangle vertices to determine polygon backface visibility and light reflection.
Step-by-Step Worked Examples in 3D Space
Calculate the cross product of a = ⟨2, 3, 4⟩ and b = ⟨5, 6, 7⟩.
1. Compute i component: (3 × 7) − (4 × 6) = 21 − 24 = -3.
2. Compute j component: −[(2 × 7) − (4 × 5)] = −[14 − 20] = −(−6) = +6.
3. Compute k component: (2 × 6) − (3 × 5) = 12 − 15 = -3.
4. Assemble vector: a × b = ⟨-3, 6, -3⟩ = -3i + 6j - 3k.
5. Magnitude (Parallelogram Area): √[(-3)² + 6² + (-3)²] = √(9 + 36 + 9) = √54 ≈ 7.348.
Result: ⟨-3, 6, -3⟩ with Parallelogram Area ≈ 7.348 square units.
Common Calculation Pitfalls & Anticommutativity Errors
The j cofactor in determinant expansion always carries an alternating negative sign: -(axbz - azbx). Missing this flips the sign of the y-component.
Unlike the dot product (a · b = b · a), the cross product is strictly anticommutative: b × a = -(a × b).
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