3D Vector Cross Product & Magnitude Calculator
Compute 3D vector cross products, orthogonal magnitudes, and angular separations.
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How to Use the Cross Product Calculator: 3D Vector Math & Magnitude
- 1Enter the 3D components for Vector A ⟨Ax, Ay, Az⟩.
- 2Enter the 3D components for Vector B ⟨Bx, By, Bz⟩.
- 3Read the resulting orthogonal cross product vector ⟨Cx, Cy, Cz⟩.
- 4Review the vector magnitude |A × B|, scalar dot product, and interior angle θ in degrees.
Formula & How It Works
3D Vector Cross Product Determinant
Calculates orthogonal vector via standard matrix determinant expansion.
Worked Examples
Cross Product of Unit Vectors i and j
Input: A = ⟨1, 0, 0⟩, B = ⟨0, 1, 0⟩
Explanation: i × j = k, pointing straight up along the positive z-axis according to the right-hand rule.
Frequently Asked Questions
What is the physical meaning of the cross product vector direction?▼
The resulting vector is strictly perpendicular (normal) to both original vectors, determined by the right-hand rule.
What happens if two vectors are parallel?▼
If two vectors are parallel or anti-parallel, their cross product is the zero vector ⟨0, 0, 0⟩ because sin(0°) = 0.
How does cross product differ from dot product?▼
Cross product produces a new 3D vector perpendicular to the inputs, while dot product produces a single scalar number.
What does the magnitude of the cross product represent geometrically?▼
The magnitude |A × B| equals the area of the parallelogram spanned by the two vectors.
Quick Reference & Conversion Table
| Vector A | Vector B | Cross Product (A × B) | Magnitude | Angle θ |
|---|---|---|---|---|
| ⟨1, 0, 0⟩ | ⟨0, 1, 0⟩ | ⟨0, 0, 1⟩ | 1.000 | 90.0° |
| ⟨0, 1, 0⟩ | ⟨0, 0, 1⟩ | ⟨1, 0, 0⟩ | 1.000 | 90.0° |
| ⟨1, 2, 3⟩ | ⟨4, 5, 6⟩ | ⟨-3, 6, -3⟩ | 7.348 | 12.9° |
| ⟨2, 0, 0⟩ | ⟨4, 0, 0⟩ | ⟨0, 0, 0⟩ | 0.000 | 0.0° (Parallel) |
⚠️ Common Pitfalls to Avoid
- •Assuming cross product is commutative (A × B = -(B × A), which points in the opposite direction).
- •Attempting to calculate cross products in 2D space without treating z as zero.
💡 Practical Real-World Uses
- •Calculating torque (τ = r × F) and angular momentum in classical mechanics.
- •Determining normal surface vectors for 3D computer graphics lighting and ray tracing.
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