What is the relation between dot product and cross product?

The dot product and cross product are two different mathematical operations that can be performed on vectors in three-dimensional Euclidean space (R^3). They serve distinct purposes and have different geometric interpretations.

Dot Product (Scalar Product):

The dot product of two vectors, often denoted as A · B or A ⋅ B, results in a scalar (a single numerical value).
It is calculated by taking the sum of the products of corresponding components of the two vectors. For two vectors A = (A1, A2, A3) and B = (B1, B2, B3), the dot product is given by:
A · B = A1 * B1 + A2 * B2 + A3 * B3.
The result of the dot product represents the cosine of the angle θ between the two vectors A and B, scaled by the magnitudes of both vectors:
A · B = |A| * |B| * cos(θ).
The dot product is commutative, meaning A · B = B · A.
It is often used to find the angle between vectors, calculate projections, and determine the similarity or orthogonality of vectors.
Cross Product (Vector Product):

The cross product of two vectors, often denoted as A × B, results in a new vector that is perpendicular to both of the original vectors.
It is calculated using the determinant of a 3x3 matrix as follows:
A × B = [(A2 * B3 - A3 * B2), (A3 * B1 - A1 * B3), (A1 * B2 - A2 * B1)].
The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors A and B, and the direction of the resulting vector follows the right-hand rule.
The cross product is not commutative; that is, A × B is not necessarily equal to B × A.
It is often used in physics and engineering to compute torque, angular momentum, and find a normal vector to a plane defined by two vectors.
In summary, the dot product produces a scalar value and is related to the angle between vectors and their magnitudes, while the cross product produces a vector that is orthogonal to the original vectors and is related to the area and orientation of the parallelogram formed by those vectors. These two operations are distinct and have different geometric interpretations and applications.