Cross Product Calculator — 3D Vector Cross Product & Right-Hand Rule
Calculate the cross product of two 3D vectors. Covers the determinant method, right-hand rule, magnitude formula, and real applications in torque and magnetic force.
The dot product gives you a number. The cross product gives you a vector — one that's perpendicular to both of the vectors you started with. It's the operation behind torque wrenches, magnetic forces, and surface normals in 3D graphics. Calculate it instantly with the CalcHub Cross Product Calculator.
The Formula
For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):
$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$
Expanding the determinant:$$\mathbf{a} \times \mathbf{b} = (a_2 b_3 - a_3 b_2)\mathbf{i} - (a_1 b_3 - a_3 b_1)\mathbf{j} + (a_1 b_2 - a_2 b_1)\mathbf{k}$$
Component by component:
- x: a₂b₃ − a₃b₂
- y: −(a₁b₃ − a₃b₁) = a₃b₁ − a₁b₃
- z: a₁b₂ − a₂b₁
Worked Example
a = (2, 3, 4), b = (5, 6, 7)- 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 of the Cross Product
$$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$$
Where θ is the angle between the vectors. This is also the area of the parallelogram spanned by a and b.
For the example above:
|a × b| = √(9 + 36 + 9) = √54 ≈ 7.35
The Right-Hand Rule
The direction of a × b follows the right-hand rule: point your fingers in the direction of a, curl them toward b, and your thumb points in the direction of a × b.
This means:
- a × b = −(b × a) — the cross product is anti-commutative
- If a and b are parallel (θ = 0° or 180°), a × b = 0 (zero vector)
Unit Vector Cross Products
| i | j | k | |
|---|---|---|---|
| i × | 0 | k | −j |
| j × | −k | 0 | i |
| k × | j | −i | 0 |
Real-World Applications
Torque: τ = r × F If you push a wrench with force F = (0, 0, −10) N at position r = (0.3, 0, 0) m from the bolt: τ = (0.3, 0, 0) × (0, 0, −10) = (0×(−10)−0×0, 0×0−0.3×(−10), 0.3×0−0×0) = (0, 3, 0) N·m The torque is 3 N·m in the y-direction. Magnetic Force: F = q(v × B) — the Lorentz force on a charged particle moving in a magnetic field. Surface Normals in 3D Graphics: Two edge vectors of a triangle, crossed together, give the face's outward normal for lighting calculations.Is the cross product defined in 2D?
The standard cross product is only defined in 3D. A 2D cross product sometimes refers to the scalar z-component of the 3D result: a₁b₂ − a₂b₁. This is useful for determining the sign of rotation (positive = counterclockwise, negative = clockwise).
Why does a × b = 0 when the vectors are parallel?
When two vectors point in the same (or opposite) direction, sin θ = 0, so the magnitude is zero. Geometrically, parallel vectors don't span a parallelogram — they define a line, and a line has zero area.