Free Cross Product Calculator with 3D Visualization

Calculate vector cross products instantly with interactive 3D visualization, step-by-step solutions, and comprehensive explanations. Perfect for physics, engineering, and mathematics students.

✅ 3D Visualization ⚡ Interactive Graph 📊 Step-by-Step 📱 Mobile Optimized

✅ Mathematically Verified

🎨 3D Visualization

📊 Interactive Controls

100%

Visual Accuracy

🧮 Interactive Cross Product Calculator

📋 What This Calculator Does

This interactive cross product calculator computes the vector product of two vectors in 3D space with real-time 3D visualization. It shows:

The resulting cross product vector in 3D space
Interactive 3D graph with rotation and zoom
Magnitude (length) of the cross product
Unit vector in the cross product direction
Angle between the original vectors
Area of the parallelogram spanned by the vectors

When to use: Physics (torque, angular momentum), Engineering (mechanics, robotics), Computer Graphics (surface normals), Mathematics (vector calculus).

Vector A (First Vector)

Vector B (Second Vector)

Quick Examples:

📈 Cross Product Results

📝 Cross Product Vector

📏 Magnitude

Length of cross product vector

🧭 Unit Vector

Direction of cross product

📐 Angle Between Vectors

Angle between A and B

🎨 Interactive 3D Visualization

Vector A

Vector B

A × B

💡 Interactive Controls: Click and drag to rotate • Scroll to zoom • Hover over vectors for details • Use control buttons above for additional options

🔬 Step-by-Step Solution

📚 Understanding Cross Product with 3D Visualization

🎯 Geometric Interpretation

The cross product produces a vector that is perpendicular to both original vectors. Its magnitude equals the area of the parallelogram spanned by the two vectors. The direction follows the right-hand rule.

|A × B| = |A| |B| sin(θ) • Direction determined by right-hand rule

🎨 How 3D Visualization Helps Understanding

Visualizing Perpendicularity

See how the cross product vector is perpendicular to both original vectors in 3D space. Rotate the view to verify from different angles.

Understanding Magnitude

The length of the cross product equals the area of the parallelogram. Compare vector lengths visually to understand the relationship.

Right-Hand Rule Visualization

Visual confirmation of the right-hand rule: curl fingers from A to B, thumb points in direction of A × B.

⚡ Real-World Applications with 3D Visualization

⚙️

Torque in Physics

τ = r × F (position vector × force)

🌀

Angular Momentum

L = r × p (position × momentum)

🎨

Surface Normals

Computer graphics lighting calculations

✈️

Aerodynamics

Lift force calculations

📝 Example Problems

Example 1: Standard Basis Vectors

Problem: Calculate i × j where i = (1,0,0) and j = (0,1,0)

Solution: i × j = k = (0,0,1)

Example 2: Parallel Vectors

Problem: Calculate A × B where A = (2,4,6) and B = (1,2,3)

Solution: A × B = (0,0,0) because vectors are parallel (B = 0.5A)

Example 3: Area of Parallelogram

Problem: Find area of parallelogram with sides A = (3,1,4) and B = (2,7,1)

Solution: Area = |A × B| = √( (-27)² + 5² + 19² ) = √(729+25+361) = √1115 ≈ 33.39

⚠️ Common Mistakes & Tips

❌ Common Mistake: Wrong Sign for j-component

Many students forget the negative sign in the j-component calculation:

Correct: A × B = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k
Wrong: A × B = (a₂b₃ - a₃b₂)i + (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k

💡 Tip: Use Determinant Method

The determinant method is less error-prone than memorizing component formulas:

✅ Best Practice: Check Orthogonality

After calculation, verify that your result is perpendicular to both original vectors using dot product:

(A × B) · A = 0 and (A × B) · B = 0

If either dot product is not zero, recheck your calculation.

❓ Cross Product & 3D Visualization FAQ

Q: How does the 3D visualization work?

A: Our 3D visualization uses Plotly.js to create interactive 3D plots. It shows Vector A (blue), Vector B (red), and their cross product A × B (green). You can rotate, zoom, and explore the vectors from any angle.

Q: Can I save or export the 3D visualization?

A: Yes! Use the "Save" button in the visualization controls to download a PNG image of the current 3D view. You can also take screenshots directly from your browser.

Q: Why is the cross product perpendicular?

A: By definition, the cross product produces a vector that is orthogonal to both input vectors. This can be verified by checking that (A × B)·A = 0 and (A × B)·B = 0.

Q: What does a zero cross product look like in 3D?

A: When vectors are parallel, their cross product is the zero vector (0,0,0). In 3D visualization, you'll see only the two original vectors lying along the same line, with no green cross product vector.