Quadratic Equation Solver (ax² + bx + c = 0)
About Cross Product Calculator
The 3D Vector Cross Product ($\vec{a} \times \vec{b}$) Calculator computes the orthogonal vector perpendicular to two 3D vectors via $3\times3$ determinant expansion, calculating right-hand rule orientation, torque vectors, and parallelogram surface areas.
How to Use Cross Product Calculator
Step 1
Enter the 3D components ($x, y, z$) for Vector A and Vector B.
Step 2
Review computed perpendicular cross product vector $\vec{a} \times \vec{b}$.
Step 3
Inspect parallelogram area and unit normal vector.
Step 4
Click "Copy Cross Product".
Practical Use Cases for Cross Product Calculator
Surface Normal Calculation for 3D Polygon Meshes
Compute surface normal vectors $(\vec{v}_1 - \vec{v}_0) \times (\vec{v}_2 - \vec{v}_0)$ for 3D triangle faces in OpenGL/DirectX rendering engines.
Physics Torque, Angular Momentum & Lorentz Force ($\vec{F} = q\vec{v} \times \vec{B}$)
Calculate magnetic Lorentz forces and mechanical rotational torque ($\vec{\tau} = \vec{r} \times \vec{F}$) with step-by-step vector arithmetic.
Input & Output Examples
Calculating Cross Product of Two 3D Vectors
Vector a: `[1, 2, 3]`, Vector b: `[4, 5, 6]`
Cross Product (a × b): `[-3, 6, -3]` | Magnitude: `7.348` (Area of Parallelogram) | Anti-Commutative (b × a): `[3, -6, 3]`
Key Features & Performance
- ✓Determinant Expansion Method: computes cross products using formal $3\times3$ matrix row expansion with unit basis vectors $\hat{i}, \hat{j}, \hat{k}$.
- ✓Calculates: Resultant Vector Components, Resultant Magnitude, Area of Parallelogram, Area of Triangle ($0.5 ||\vec{a} \times \vec{b}||$).
- ✓Interactive 3D visualizer showing the perpendicular right-hand normal vector.
- ✓100% Client-Side 3D math engine.
- ✓1-Click Copy resultant vector.
Key Terminology & Definitions
Vector Cross Product ($\vec{a} \times \vec{b}$)
A binary operation on two vectors in three-dimensional space resulting in a third vector perpendicular to both original vectors.
Right-Hand Rule
A spatial orientation convention stating that if the right hand fingers curl from $\vec{a}$ to $\vec{b}$, the thumb points in the direction of $\vec{a} \times \vec{b}$.