What is the Difference Between Dot Product and Cross Product?

The main difference between the dot product and the cross product of two vectors lies in the nature of the result they produce and their geometric interpretation. Here are the key differences:

1. Result: The dot product results in a scalar quantity, which indicates magnitude but not direction, while the cross product results in a vector quantity, which indicates both magnitude and direction.
2. Geometric Interpretation: The dot product measures the degree of parallelism between two vectors, ranging from 0 (perpendicular vectors) to the product of the lengths of the two vectors (parallel vectors). The cross product, on the other hand, generates a vector that is perpendicular (orthogonal) to the plane created by the two input vectors.
3. Applications: Dot product is used to calculate projections, cosines of angles, and inner products, while cross product is used to calculate normal vectors, torque, and angles between vectors with non-zero magnitudes.

In summary, the dot product is a scalar that measures the degree of parallelism between two vectors, while the cross product is a vector that spans the perpendicular plane between the same two vectors.

Comparative Table: Dot Product vs Cross Product

The dot product and cross product are two mathematical operations performed on vectors. Here is a table summarizing the differences between them:

The dot product is defined between any two vectors of the same size (but arbitrary), and it results in a scalar quantity, which indicates magnitude but not direction. On the other hand, the cross product is defined only for 3D vectors and results in a vector quantity, which indicates both magnitude and direction.