Dot and Cross Product Visualization
The dot product gives a number, the cross product gives a vector, and it is easy to lose track of which is which. This 3D simulation lets you drag two vectors around and watch both products respond in real time.
How to use this simulation
- Drag either vector to any orientation in 3D
- Change the vector magnitudes
- Display the resulting scalar and vector products
What to look for
- The dot product falls to zero at 90° and is negative beyond it
- The cross product is largest at 90° and zero when the vectors line up
- Swapping the order of a cross product reverses the result
The physics behind it
The dot product a·b = |a||b|cosθ measures how much one vector lies along the other, and it vanishes when they are perpendicular. The cross product a×b has magnitude |a||b|sinθ and points perpendicular to both by the right-hand rule; it vanishes when the vectors are parallel. Work uses the dot product, torque and magnetic force use the cross product.