Dot and Cross Product Calculator

Calculate dot and cross products with geometric interpretations and 2D/3D visualizations

Overview

The Dot and Cross Product Calculator computes vector products and provides geometric interpretations through interactive visualizations. The dot product measures how much two vectors point in the same direction (producing a scalar), while the cross product finds a vector perpendicular to both input vectors (3D only), with both operations having fundamental geometric and physical meanings.

Tips

  1. Understand the Sign of Dot Products: A positive dot product means vectors point in generally the same direction, zero means they’re perpendicular, and negative means they point in opposite directions.

  2. Use the Right-Hand Rule for Cross Products: Point your fingers along the first vector, curl them toward the second vector, and your thumb points in the direction of the cross product - this is essential for understanding 3D orientations.

  3. Try Perpendicular and Parallel Vectors: Test vectors at right angles (dot product should be zero) and parallel vectors (cross product should be zero) to verify the fundamental properties.

  4. Visualize Angles and Projections: Use the 2D visualization to see how the dot product relates to the angle between vectors and how one vector projects onto another.

  5. Remember Cross Products Are 3D Only: While dot products work in any dimension, cross products as traditionally defined only work in 3D space (in 2D, the result is a scalar representing the z-component).