Vector Cross Product

Calculate the cross product of two 3D vectors online quickly and easily. Get step-by-step solutions with this free vector cross product calculator.

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About Vector Cross Product Tool

The Vector Cross Product Tool ✨🧮 is a precise, easy-to-use calculator that helps you find the cross product between two 3D vectors, showing both the magnitude and direction of the resulting vector.

Whether you’re working on physics problems, computer graphics, or engineering mechanics, this tool provides instant step-by-step solutions, complete with formulas, unit vector breakdowns, and visual 3D orientation diagrams.

It’s ideal for students, engineers, developers, and researchers who want to understand the geometric meaning of the vector product — a perpendicular vector representing rotational or area properties.

⚙️ Key Features:

🧮 Instant Cross Product Calculation:
Computes the vector cross product using the standard determinant formula:
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A × B = (AyBz – AzBy)i – (AxBz – AzBx)j + (AxBy – AyBx)k
Where:
- A = (Ax, Ay, Az)
- B = (Bx, By, Bz)
- Result = C = (Cx, Cy, Cz)
📊 Step-by-Step Solution Display:
Shows every stage of computation clearly:
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A = (2, 3, 4), B = (5, 6, 7) Cx = (3×7 – 4×6) = –3 Cy = –(2×7 – 4×5) = 6 Cz = (2×6 – 3×5) = –3 Result: C = (–3, 6, –3)
🧠 Magnitude Calculation:
Automatically computes the magnitude of the resulting vector:
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|A × B| = √(Cx² + Cy² + Cz²)
✅ Example:
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|A × B| = √((-3)² + 6² + (-3)²) = 7.35
⚡ Angle Finder Between Vectors:
Displays the sine-based relationship used in vector multiplication:
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|A × B| = |A| × |B| × sin(θ)
and computes θ automatically if magnitudes are known.
📐 Geometric Visualization (3D Plot):
Interactive 3D diagram showing:
- Vector A →
- Vector B →
- Resultant A×B ⊥ (perpendicular to both)
🧭 Direction & Orientation Info:
Uses the right-hand rule 🖐️ to show orientation of the resultant vector.
🔢 Unit Vector Option:
Normalize vectors for direction-only analysis:
```
 
```
û = (A × B) / |A × B|
🧩 Supports Multiple Units & Contexts:
Works for all vector applications — whether units are in N (Newtons), m (meters), A (amperes), or unitless dimensions.
📏 Additional Vector Tools:
Optionally switch to:
- Dot Product Calculator (scalar)
- Triple Product (A · (B × C))
- Projection and Angle Calculators
📱 Responsive & Visual:
Works perfectly across desktop, tablet, and mobile with clean, educational visualization.
🔒 Private & Secure:
100% offline computation — your data stays local.

💡 How It Works (Simplified):

The cross product of two 3D vectors produces a third vector that is perpendicular to both of the original vectors.
Its magnitude represents the area of the parallelogram spanned by A and B.

🧮 Formula:

A × B = |A| × |B| × sin(θ) × n̂

Where:

θ = angle between A and B
n̂ = unit vector perpendicular to both A and B

Component Form (Determinant):

A × B = | i j k | | Ax Ay Az | | Bx By Bz |

📘 Example Calculations:

Example 1️⃣ – Simple Cross Product

A = (2, 3, 4), B = (5, 6, 7) A × B = (–3, 6, –3) |A × B| = √(54) = 7.35

✅ Result: Perpendicular vector = (–3, 6, –3)
✅ Magnitude: 7.35

Example 2️⃣ – Perpendicular Vectors

A = (1, 0, 0), B = (0, 1, 0) A × B = (0, 0, 1)

✅ Result: The vectors are perpendicular; cross product points along the z-axis.

Example 3️⃣ – Parallel Vectors

A = (2, 4, 6), B = (1, 2, 3) A × B = (0, 0, 0)

✅ Result: Zero vector → No perpendicular direction (vectors are parallel).

🧭 Perfect For:

⚙️ Engineers: Analyze torque, rotational motion, or moment of force.
🧮 Mathematicians: Study vector spaces, determinants, and geometry.
🧲 Physicists: Compute magnetic forces (F = qv × B).
🎮 Game Developers: Handle 3D rendering, surface normals, and camera rotation.
🎓 Students: Understand cross product direction and area relations visually.

🔍 Why It’s Valuable:

The Vector Cross Product Tool transforms complex 3D math into clear, intuitive results.

It helps users:
✅ Quickly compute perpendicular vectors.
✅ Understand spatial orientation with 3D visuals.
✅ Verify work in physics, CAD, or graphics systems.
✅ Learn geometric significance through live examples.
✅ Reinforce right-hand rule understanding.

It’s not just a calculator — it’s your visual guide to 3D vector geometry.

🧩 Advanced Options (Optional):

📈 Triple Product Visualization: See how A·(B×C) represents volume.
🧾 Angle Display Mode: Compute both sin(θ) and cos(θ) for hybrid analysis.
🧱 Vector Projection Add-on: Show Aₙ and Aₜ (normal and tangential components).
⚙️ Coordinate Conversion: Work in Cartesian, cylindrical, or spherical systems.
🎛️ 3D Animation Mode: Rotate the 3D model to see vector relationships dynamically.

🌍 Common Use Cases:

Application	Input	Output	Result Meaning
Torque	r × F	(τx, τy, τz)	Moment of force
Magnetic Force	q(v × B)	(Fx, Fy, Fz)	Lorentz force
Surface Normal	AB × AC	(nx, ny, nz)	Normal to surface
Geometry	A × B	Perpendicular vector	Plane orientation
3D Graphics	Axis × Rotation	(vx, vy, vz)	Orientation vector

🧠 Scientific Insight:

The cross product represents rotational direction and area magnitude — it’s foundational in 3D physics and geometry.

It defines torque, angular momentum, and magnetic force.
It’s geometrically the area vector of the parallelogram formed by A and B.
It’s always orthogonal to both A and B, determined by the right-hand rule.

This makes it indispensable in mechanics, electromagnetism, robotics, and 3D modeling.

✨ In Short:

The Vector Cross Product Tool 🧭⚙️ transforms vector algebra into a visual, educational, and computational experience.
It gives you the direction, magnitude, and geometric meaning of 3D vector interactions — instantly and clearly.

Calculate. Visualize. Understand.
With the Vector Cross Product Calculator, 3D math becomes visual, intuitive, and powerful. 📈💡🌐